1. Calculus and Analytical Geometry Lecture # 8 MTH 104

2. 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

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

4. Techniques of differentiation

5. 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

6. Techniques of differentiation

7. 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

8. Techniques of differentiation

9. Example Find dy/dx if solution

10. 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

11. 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

12. Techniques of differentiation Example Find dy/dx, if solution

13. 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

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

15. Techniques of differentiation

16. Higher order derivatives If y=f(x) then

17. Higher order derivatives A general nth order derivative

18. Example Solution First Order derivat ive Second order derivative

19. Third order derivative

20. Example Find Solution

21. Derivative of trigonometric functions

22. Example Solution

23. Example solution

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

25. Example Given thatshow that