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Tangent and Derivative I

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LIMITS In order to understand differentiability and continuity, we must be familiar with the limit notation.

Limit Notation means as

approaches ,

approaches or goes as

close as we like to For simple limits, we can simply substitute the appropriate value of

and evaluate

Question 1 Evaluate the following limits a)

b)

c)

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BASIC OPERATIONS AND LIMITS These theorems need to be known but not proven. They are intuitively obvious.

Limit Theorems

Question 2 Evaluate the following limits a)

b)

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c)

d)

Talent Tip: When doing limit questions, you do not have to explicitly split each limit into smaller limits and then evaluate, as we have done here. Substituting the values directly is enough. However, you must always explicitly show when you substitute your value into the limit, and not evaluate further. E.g. in c), do not directly evaluate

as , but write in

the substitution step. This is so examiners know you understand the limit, and are not doing it on a calculator.

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Tangent and Derivative I

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FINDING LIMITS OF POLYNOMIAL FUNCTIONS While some limits are simple substitutions, other limits will require manipulation before they can be evaluated. Mostly, these involve polynomial functions, and in particular, fractions

Limit as

of

Note that this applies to all powers of as well

Finding Limits of Polynomial Functions When the denominator Factorise the numerator and denominator, cancel any common factors, and then evaluate When Divide the top and bottom by the highest power of

in the numerator,

and use the limit of above

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Question 3 Evaluate the following limits a)

b)

c)

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d)

e)

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Tangent and Derivative I

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GEOMETRIC DEFINITION OF ‘DERIVATIVE’ The derivative of a function is denoted

:

Definition:

SLOPE OF A CURVE The gradient is the slope of the curve (or the slope of the tangent to the curve)

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Tangent and Derivative I

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HORIZONTAL TANGENT Question 4 – Horizontal Tangent (Conceptual) Consider the two functions below, in which the curves are becoming progressively flatter

a) Calculate the gradient of each of the curves. Rise of the curve

units

Run of the curve

units

Gradient

(First curve)

Rise of curve

units

Gradient

(Second Curve)

Gradient

(Third curve)

b) Hence, explain what happens to the value of the gradient as the curve becomes flatter? As the curve becomes flatter, the value of the rise decreases to , while the run stays the same. Hence the gradient will get closer and closer to c) What would the value of the gradient be when the curve is completely horizontal? [HINT:

]

When the curve is horizontal, the tangent will be horizontal The rise will be Gradient TALENT 100: HSC SUCCESS. SIMPLIFIED. Page 9 of 46

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VERTICAL TANGENT Question 5 – Horizontal Tangent (Conceptual) Consider the two functions below, in which the curves are becoming progressively steeper.

a) Calculate the gradient of each of the curves. Rise of the curve

units

Run of the curve

units

Gradient

(First curve)

Run of curve

units

Gradient

(Second Curve)

Gradient

(Third curve)

b) Hence, explain what happens to the value of the gradient as the curve becomes steeper? As the curve becomes flatter, the value of the run decreases to , while the run stays the same. Hence the gradient will get closer and closer to infinity c) What would the value of the gradient be when the curve becomes vertical? [HINT: ] When the curve is vertical, the tangent is vertical The rise will be

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DIFFERENTIATION FROM FIRST PRINCIPLES The gradient of the function is equal to the slope:

In the following diagram, P and Q are two points on the curve coordinates

and hence have

and

Rise of PQ

…………………………………………..

Run of PQ

………………………………………….. …………………………………………..

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Tangent and Derivative I

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Gradient of Secant

Gradient of Tangent We can find the gradient of the tangent, by considering what happens when Q moves closer and closer to P, or when

Differentiation from First Principles

Talent Tip: When differentiating from first principles, we seek to eliminate the h’s from the denominator. The questions will illustrate.

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Question 6 Find the derivative of the following functions using first principles a) Let

b) Let

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[HINT: rationalise the numerator] Let

d) Let

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THE NOTATION

The graph above shows a point

, and another point Q that is a small distance away

from P. We denote a small change by the sign . Hence, Q has co-ordinates )

For a very small

(i.e.

, the gradient of the secant becomes the gradient of the

tangent. We define:

Different notations of the derivative For a function

, the derivative can be written as

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Tangent and Derivative I

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DIFFERENTIATING POLYNOMIAL FUNCTIONS DIFFERENTIATING Differentiating Powers of

Talent Tip: An easy way to consider this principle is that you “bring down” the power, and then minus one from it

Question 7 Differentiate a)

b)

c)

d)

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Question 8 Differentiate a)

b)

c)

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Question 9 Differentiate the following – a)

b)

c)

d)

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Tangent and Derivative I

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BASIC OPERATIONS Just like there are limit theorems, there are also ones for derivatives. This is not surprising as the definition of the derivative comes from limits. However, note that only the first two laws apply, and not multiplication or division.

Derivative Theorems

Talent Tip: Note that the derivative of a constant is

Question 10 Differentiate the following a)

b)

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c)

d)

e)

f)

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Question 11 Differentiate the following functions a)

b)

c)

d)

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e)

f)

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Tangent and Derivative I

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FINDING THE EQUATION OF TANGENTS AND NORMALS Finding the Tangent/Normal STEP 1: Find the

co-ordinate of the point

STEP 2: Find the derivative

, and then find the gradient of the tangent

or normal (Remember STEP 3: Find the equation of the line using point-gradient form Question 1 Find the equation of the tangent to curve STEP 1: Find the

at

co-ordinate of the point

So the co-ordinates of the point are STEP 2: Find the derivative

, and then find the gradient of the tangent or normal

STEP 3: Now use point-gradient form to find the equation of the tangent

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Question 2 Find the equation of the tangent to the curve STEP 1: Find the

at the point

co-ordinate of the point

The point is

STEP 2: Find the derivative

, and then find the gradient of the tangent or normal

STEP 3: Now use point-gradient form to find the equation of the tangent

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Tangent and Derivative I

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Question 3 Find the equation of the tangent to the curve STEP 1: Find the

at

co-ordinate of the point

So the co-ordinates of the point are

STEP 2: Find the derivative

, and then find the gradient of the tangent or normal

STEP 3: Now use point-gradient form to find the equation of the tangent

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Question 4 Consider the graph of

. The tangent to the curve at the point

intersects the

and

and -axis at the points

respectively

a) Find the equation of the tangent

When

,

Gradient of tangent Equation of tangent:

b) Find the area of the triangle

, where

is the origin

When

When

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THE CHAIN RULE To differentiate functions such as

, we need to recognize that the function is

composed of a chain of two functions

and

that we can differentiate

separately: Square and add one

Cube the new function

We use the chain rule to see how to differentiate the combined function

The Chain Rule

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Question 5 a) Differentiate

with respect to

b) Hence, using the chain rule, differentiate ..................... ............................ .........................

.........................

Question 6 a) Differentiate

with respect to

b) Hence, using the chain rule, differentiate ..................... ............................ .........................

.........................

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Question 7 Differentiate the following functions using the chain rule a) ..................... ............................ .........................

.........................

b) ..................... ............................ .........................

.........................

c) ..................... ............................ .........................

.........................

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d) ..................... ............................ .........................

.........................

e) ..................... ............................ .........................

.........................

f) ..................... ............................ .........................

.........................

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g) ..................... ............................ .........................

.........................

h) ..................... ............................ .........................

.........................

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Tangent and Derivative I

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LINEAR FUNCTIONS Differentiating Linear Functions

Talent Tip: “Take the derivative of the OUTSIDE of brackets times by the derivative of inside the brackets”

Question 8 Find the derivative of the following a)

b)

c)

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d)

e)

f)

g)

h)

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THE PRODUCT RULE The Product Rule Suppose we have a function

, where

and

are simpler functions

Talent Tip: You will often be required to use both the chain rule and the product rule Talent Tip: Unless the question specifies otherwise, you do not have to factorise the derivative.

Question 9 a) Using the chain rule, differentiate

b) Using the product rule, differentiate ................................. ....................................... ................................. ......................................

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Question 10 a) Use the chain rule to differentiate

b) Hence, find the derivative of ................................. ....................................... ..............................… ......................................

Question 11 a) Find

and

where

and

b) Hence differentiate ................................. ....................................... ................................. ......................................

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Question 12 Use the product rule to differentiate the following with respect to a) ................................. ....................................... ................................. ...................................... b) ................................. ....................................... ................................. ......................................

c) ................................. ....................................... ................................

d) ................................. ....................................... ................................

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Question 13 Differentiate and then factorise the following a)

b)

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Question 14 Consider the function a) Find

b) At what -values is the gradient of the tangent perpendicular to the gradient at

?

When

The tangent is perpendicular to the tangent at

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at

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THE QUOTIENT RULE The Quotient Rule Suppose we have a function

, where

and

are simpler functions

The order of differentiation is important Question 15 Differentiate a) Find

and

using the quotient rule when

b) Hence, differentiate ................................. ....................................... ................................. ......................................

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using the product rule, simplifying your answer ................................. ....................................... ................................. ......................................

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Question 16 Use the quotient rule to find the derivatives of the following functions a) ................................. ....................................... ................................. ......................................

b) ................................. ....................................... ................................. ......................................

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c) ................................. ....................................... ................................. ......................................

d) ................................. ....................................... ................................. ......................................

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Question 17 Find the derivatives of the following functions, and find the value(s) of

for which the

gradient of the tangent is equal to a)

Gradient of the tangent is

when

Hence, there are no values of

for which the gradient of the tangent is

b)

Gradient of the tangent is

when

Hence, there are no real values of

for which the gradient is

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Question 18 Consider the function a) If

, find

b) For what values of

is the gradient positive? Negative? Zero?

Gradient positive when i.e.

since

for all

excluding Gradient zero when

Gradient negative when

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Question 19 Consider the curve a) Show that the point When

lies on the curve

and

Hence, the point

lies on the curve

b) Using the quotient rule, find

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c) Hence, find the equation of the normal at the point When

Equation of Normal:

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Tangent and Derivative I

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LIMITS In order to understand differentiability and continuity, we must be familiar with the limit notation.

Limit Notation means as

approaches ,

approaches or goes as

close as we like to For simple limits, we can simply substitute the appropriate value of

and evaluate

Question 1 Evaluate the following limits a)

b)

c)

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Tangent and Derivative I

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BASIC OPERATIONS AND LIMITS These theorems need to be known but not proven. They are intuitively obvious.

Limit Theorems

Question 2 Evaluate the following limits a)

b)

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Tangent and Derivative I

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c)

d)

Talent Tip: When doing limit questions, you do not have to explicitly split each limit into smaller limits and then evaluate, as we have done here. Substituting the values directly is enough. However, you must always explicitly show when you substitute your value into the limit, and not evaluate further. E.g. in c), do not directly evaluate

as , but write in

the substitution step. This is so examiners know you understand the limit, and are not doing it on a calculator.

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Tangent and Derivative I

© TALENT 100

FINDING LIMITS OF POLYNOMIAL FUNCTIONS While some limits are simple substitutions, other limits will require manipulation before they can be evaluated. Mostly, these involve polynomial functions, and in particular, fractions

Limit as

of

Note that this applies to all powers of as well

Finding Limits of Polynomial Functions When the denominator Factorise the numerator and denominator, cancel any common factors, and then evaluate When Divide the top and bottom by the highest power of

in the numerator,

and use the limit of above

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Tangent and Derivative I

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Question 3 Evaluate the following limits a)

b)

c)

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d)

e)

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Tangent and Derivative I

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GEOMETRIC DEFINITION OF ‘DERIVATIVE’ The derivative of a function is denoted

:

Definition:

SLOPE OF A CURVE The gradient is the slope of the curve (or the slope of the tangent to the curve)

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Tangent and Derivative I

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HORIZONTAL TANGENT Question 4 – Horizontal Tangent (Conceptual) Consider the two functions below, in which the curves are becoming progressively flatter

a) Calculate the gradient of each of the curves. Rise of the curve

units

Run of the curve

units

Gradient

(First curve)

Rise of curve

units

Gradient

(Second Curve)

Gradient

(Third curve)

b) Hence, explain what happens to the value of the gradient as the curve becomes flatter? As the curve becomes flatter, the value of the rise decreases to , while the run stays the same. Hence the gradient will get closer and closer to c) What would the value of the gradient be when the curve is completely horizontal? [HINT:

]

When the curve is horizontal, the tangent will be horizontal The rise will be Gradient TALENT 100: HSC SUCCESS. SIMPLIFIED. Page 9 of 46

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Tangent and Derivative I

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VERTICAL TANGENT Question 5 – Horizontal Tangent (Conceptual) Consider the two functions below, in which the curves are becoming progressively steeper.

a) Calculate the gradient of each of the curves. Rise of the curve

units

Run of the curve

units

Gradient

(First curve)

Run of curve

units

Gradient

(Second Curve)

Gradient

(Third curve)

b) Hence, explain what happens to the value of the gradient as the curve becomes steeper? As the curve becomes flatter, the value of the run decreases to , while the run stays the same. Hence the gradient will get closer and closer to infinity c) What would the value of the gradient be when the curve becomes vertical? [HINT: ] When the curve is vertical, the tangent is vertical The rise will be

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Tangent and Derivative I

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DIFFERENTIATION FROM FIRST PRINCIPLES The gradient of the function is equal to the slope:

In the following diagram, P and Q are two points on the curve coordinates

and hence have

and

Rise of PQ

…………………………………………..

Run of PQ

………………………………………….. …………………………………………..

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Tangent and Derivative I

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Gradient of Secant

Gradient of Tangent We can find the gradient of the tangent, by considering what happens when Q moves closer and closer to P, or when

Differentiation from First Principles

Talent Tip: When differentiating from first principles, we seek to eliminate the h’s from the denominator. The questions will illustrate.

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Tangent and Derivative I

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Question 6 Find the derivative of the following functions using first principles a) Let

b) Let

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[HINT: rationalise the numerator] Let

d) Let

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Tangent and Derivative I

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THE NOTATION

The graph above shows a point

, and another point Q that is a small distance away

from P. We denote a small change by the sign . Hence, Q has co-ordinates )

For a very small

(i.e.

, the gradient of the secant becomes the gradient of the

tangent. We define:

Different notations of the derivative For a function

, the derivative can be written as

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PMX

Tangent and Derivative I

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DIFFERENTIATING POLYNOMIAL FUNCTIONS DIFFERENTIATING Differentiating Powers of

Talent Tip: An easy way to consider this principle is that you “bring down” the power, and then minus one from it

Question 7 Differentiate a)

b)

c)

d)

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Question 8 Differentiate a)

b)

c)

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Question 9 Differentiate the following – a)

b)

c)

d)

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Tangent and Derivative I

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BASIC OPERATIONS Just like there are limit theorems, there are also ones for derivatives. This is not surprising as the definition of the derivative comes from limits. However, note that only the first two laws apply, and not multiplication or division.

Derivative Theorems

Talent Tip: Note that the derivative of a constant is

Question 10 Differentiate the following a)

b)

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c)

d)

e)

f)

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Question 11 Differentiate the following functions a)

b)

c)

d)

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e)

f)

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FINDING THE EQUATION OF TANGENTS AND NORMALS Finding the Tangent/Normal STEP 1: Find the

co-ordinate of the point

STEP 2: Find the derivative

, and then find the gradient of the tangent

or normal (Remember STEP 3: Find the equation of the line using point-gradient form Question 1 Find the equation of the tangent to curve STEP 1: Find the

at

co-ordinate of the point

So the co-ordinates of the point are STEP 2: Find the derivative

, and then find the gradient of the tangent or normal

STEP 3: Now use point-gradient form to find the equation of the tangent

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Question 2 Find the equation of the tangent to the curve STEP 1: Find the

at the point

co-ordinate of the point

The point is

STEP 2: Find the derivative

, and then find the gradient of the tangent or normal

STEP 3: Now use point-gradient form to find the equation of the tangent

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Question 3 Find the equation of the tangent to the curve STEP 1: Find the

at

co-ordinate of the point

So the co-ordinates of the point are

STEP 2: Find the derivative

, and then find the gradient of the tangent or normal

STEP 3: Now use point-gradient form to find the equation of the tangent

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Question 4 Consider the graph of

. The tangent to the curve at the point

intersects the

and

and -axis at the points

respectively

a) Find the equation of the tangent

When

,

Gradient of tangent Equation of tangent:

b) Find the area of the triangle

, where

is the origin

When

When

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THE CHAIN RULE To differentiate functions such as

, we need to recognize that the function is

composed of a chain of two functions

and

that we can differentiate

separately: Square and add one

Cube the new function

We use the chain rule to see how to differentiate the combined function

The Chain Rule

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Question 5 a) Differentiate

with respect to

b) Hence, using the chain rule, differentiate ..................... ............................ .........................

.........................

Question 6 a) Differentiate

with respect to

b) Hence, using the chain rule, differentiate ..................... ............................ .........................

.........................

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Question 7 Differentiate the following functions using the chain rule a) ..................... ............................ .........................

.........................

b) ..................... ............................ .........................

.........................

c) ..................... ............................ .........................

.........................

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d) ..................... ............................ .........................

.........................

e) ..................... ............................ .........................

.........................

f) ..................... ............................ .........................

.........................

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g) ..................... ............................ .........................

.........................

h) ..................... ............................ .........................

.........................

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LINEAR FUNCTIONS Differentiating Linear Functions

Talent Tip: “Take the derivative of the OUTSIDE of brackets times by the derivative of inside the brackets”

Question 8 Find the derivative of the following a)

b)

c)

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d)

e)

f)

g)

h)

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THE PRODUCT RULE The Product Rule Suppose we have a function

, where

and

are simpler functions

Talent Tip: You will often be required to use both the chain rule and the product rule Talent Tip: Unless the question specifies otherwise, you do not have to factorise the derivative.

Question 9 a) Using the chain rule, differentiate

b) Using the product rule, differentiate ................................. ....................................... ................................. ......................................

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Question 10 a) Use the chain rule to differentiate

b) Hence, find the derivative of ................................. ....................................... ..............................… ......................................

Question 11 a) Find

and

where

and

b) Hence differentiate ................................. ....................................... ................................. ......................................

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Question 12 Use the product rule to differentiate the following with respect to a) ................................. ....................................... ................................. ...................................... b) ................................. ....................................... ................................. ......................................

c) ................................. ....................................... ................................

d) ................................. ....................................... ................................

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Question 13 Differentiate and then factorise the following a)

b)

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Question 14 Consider the function a) Find

b) At what -values is the gradient of the tangent perpendicular to the gradient at

?

When

The tangent is perpendicular to the tangent at

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at

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THE QUOTIENT RULE The Quotient Rule Suppose we have a function

, where

and

are simpler functions

The order of differentiation is important Question 15 Differentiate a) Find

and

using the quotient rule when

b) Hence, differentiate ................................. ....................................... ................................. ......................................

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using the product rule, simplifying your answer ................................. ....................................... ................................. ......................................

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Question 16 Use the quotient rule to find the derivatives of the following functions a) ................................. ....................................... ................................. ......................................

b) ................................. ....................................... ................................. ......................................

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c) ................................. ....................................... ................................. ......................................

d) ................................. ....................................... ................................. ......................................

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Question 17 Find the derivatives of the following functions, and find the value(s) of

for which the

gradient of the tangent is equal to a)

Gradient of the tangent is

when

Hence, there are no values of

for which the gradient of the tangent is

b)

Gradient of the tangent is

when

Hence, there are no real values of

for which the gradient is

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Question 18 Consider the function a) If

, find

b) For what values of

is the gradient positive? Negative? Zero?

Gradient positive when i.e.

since

for all

excluding Gradient zero when

Gradient negative when

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Question 19 Consider the curve a) Show that the point When

lies on the curve

and

Hence, the point

lies on the curve

b) Using the quotient rule, find

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c) Hence, find the equation of the normal at the point When

Equation of Normal:

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