Calculus and Analytical Geometry Lecture # 8 MTH 104

Techniques of differentiation 1. Constant Function Rule: The derivative of a constant function is zero. y = f(x) = c where c is a constant Examples

Techniques of differentiation 2. Power Rule: Let, where the dependant variable x is raised to a constant value, the power n, then Examples

Techniques of differentiation

3. Constant Multiplied by a Function Rule: Let y be equal to the product of a constant c and some function f(x), such that y = cf(x) then Techniques of differentiation Examples

Techniques of differentiation

4. Sum (Difference) Rule: Let y be the sum (difference) of two functions (differentiable) f(x) and g(x). y = f(x) + g(x ), then Examples

Techniques of differentiation

Example Find dy/dx if solution

Techniques of differentiation Example At what points, if any does the graph of have a horizontal tangent line? solution Slope of horizontal line is zero that is dy/dx=0

Techniques of differentiation 4. Product Rule: Let y = f(x).g(x), where f(x) and g(x) are two differentiable functions of the variable x. Then

Techniques of differentiation Example Find dy/dx, if solution

Techniques of differentiation 5. Quotient Rule: Let y = f(x)/g(x), where f(x) and g(x) are two differentiable functions of the variable x and g(x) ≠ 0. Then

Techniques of differentiation Example Find dy/dx if solution Derivative of numerator Derivative of denominator

Techniques of differentiation

Higher order derivatives If y=f(x) then

Higher order derivatives A general nth order derivative

Example Solution First Order derivat ive Second order derivative

Third order derivative

Example Find Solution

Derivative of trigonometric functions

Example Solution

Example solution

Substituting the valuse ofinto (1) L.H.S=R.H.S

Example Given thatshow that