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DIFFERENTIAL CALCULUS (4)

I. CONCEPTS, DEFINITIONS, THEOREMS AND FORMULAS A. THE DERIVATIVE a.1. Definition: Let y = f(x). If ∆x is any increment given to the independent variable x and ∆y is the corresponding increment in the dependent variable y, then the derivative of the function, y = f(x), with respect to x is the limit of the ratio of ∆y to ∆x as ∆x approaches zero, or

b.3.

, a composite function, then

(5)

(1) C. SOME IMPORTANT FORMULAS ON DIFFERENTIATION a.2. Higher Derivatives Eq’n. (1) represents the first derivative of y relative to x. The derivatives higher than the first are defined as follows:

1.

where k is any constant

2.

(a) second derivative: 3. (b) third derivative

where u, v, … are functions of x

: 4.

General Formula to get the nth derivative (2) B. RELATIONS AMONG THE DERIVATIVES

b.1.

b.2. with t as the parameter, then

(3)

which is known as a parametric equation

5.

6. C.1. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 1.

2.

3.

4.

C.4. DERIVATIVES OF EXPONENTIAL FUNCTIONS 1.

5. 2. 6. C.5. DERIVATIVES OF VARIABLE WITH VARIABLE EXPONENTS C.2. DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS 1. 1.

2.

3.

C.6. DERIVATIVES OF HYPERBOLIC FUNCTIONS 1.

2.

3. 4. D. SLOPE OF A CURVE: TANGENT AND NORMAL LINES 5. C.2. DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS

6. C.3. DERIVATIVES OF LOGARITHMIC FUNCTIONS

1. The derivative

of the function y = f(x) geometrically represents

the slope of the tangent line at any point on the curve defined by y = f(x), or (6) 2. The equation of the tangent line on y = f(x) at the point P1(x1, y1), by the point-slope form, is

1. 3. The equation of the line normal to the curve at the point P1(x1, y1) is 2.

3.

1.

The maximum/minimum value of the function y = f(x), is the value of the function which Is larger/smaller than the values of the function in its immediate vicinity or neighborhood.

2. Test for a Maximum at the point x = x1: f’(x1) = 0

and

f’’(x1) < 0.

3. Test for a Minimum at the point x = x1: f’(x1) = 0

and

f’’(x1) > 0.

NOTE: Maximum and minimum values of a function are values for which the derivative of the function,

E. DERIVATIVE OF ARC LENGTH, RADIUS OF CURVATURE 1. If s represent the arc length measured along the curve y = f(x), then

minimum) is called the critical value of the function. G. POINT OF INFLECTION

(7)

(b)

(8)

(c)

(9)

2. Radius of Curvature, ρ: the radius of the circle that is tangent to the curve y= f(x) at any point and with center on the concave side of this curve.

(10)

3. Curvature, K: the reciprocal of the radius of curvature (11) F. MAXIMUM AND MINIMUM VALUES OF A FUNCTION

Either value (maximum or

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I. CONCEPTS, DEFINITIONS, THEOREMS AND FORMULAS A. THE DERIVATIVE a.1. Definition: Let y = f(x). If ∆x is any increment given to the independent variable x and ∆y is the corresponding increment in the dependent variable y, then the derivative of the function, y = f(x), with respect to x is the limit of the ratio of ∆y to ∆x as ∆x approaches zero, or

b.3.

, a composite function, then

(5)

(1) C. SOME IMPORTANT FORMULAS ON DIFFERENTIATION a.2. Higher Derivatives Eq’n. (1) represents the first derivative of y relative to x. The derivatives higher than the first are defined as follows:

1.

where k is any constant

2.

(a) second derivative: 3. (b) third derivative

where u, v, … are functions of x

: 4.

General Formula to get the nth derivative (2) B. RELATIONS AMONG THE DERIVATIVES

b.1.

b.2. with t as the parameter, then

(3)

which is known as a parametric equation

5.

6. C.1. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 1.

2.

3.

4.

C.4. DERIVATIVES OF EXPONENTIAL FUNCTIONS 1.

5. 2. 6. C.5. DERIVATIVES OF VARIABLE WITH VARIABLE EXPONENTS C.2. DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS 1. 1.

2.

3.

C.6. DERIVATIVES OF HYPERBOLIC FUNCTIONS 1.

2.

3. 4. D. SLOPE OF A CURVE: TANGENT AND NORMAL LINES 5. C.2. DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS

6. C.3. DERIVATIVES OF LOGARITHMIC FUNCTIONS

1. The derivative

of the function y = f(x) geometrically represents

the slope of the tangent line at any point on the curve defined by y = f(x), or (6) 2. The equation of the tangent line on y = f(x) at the point P1(x1, y1), by the point-slope form, is

1. 3. The equation of the line normal to the curve at the point P1(x1, y1) is 2.

3.

1.

The maximum/minimum value of the function y = f(x), is the value of the function which Is larger/smaller than the values of the function in its immediate vicinity or neighborhood.

2. Test for a Maximum at the point x = x1: f’(x1) = 0

and

f’’(x1) < 0.

3. Test for a Minimum at the point x = x1: f’(x1) = 0

and

f’’(x1) > 0.

NOTE: Maximum and minimum values of a function are values for which the derivative of the function,

E. DERIVATIVE OF ARC LENGTH, RADIUS OF CURVATURE 1. If s represent the arc length measured along the curve y = f(x), then

minimum) is called the critical value of the function. G. POINT OF INFLECTION

(7)

(b)

(8)

(c)

(9)

2. Radius of Curvature, ρ: the radius of the circle that is tangent to the curve y= f(x) at any point and with center on the concave side of this curve.

(10)

3. Curvature, K: the reciprocal of the radius of curvature (11) F. MAXIMUM AND MINIMUM VALUES OF A FUNCTION

Either value (maximum or

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