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Calculus and Analytical Geometry

MTH 104. Calculus and Analytical Geometry. Lecture # 8. 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 :

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Calculus and Analytical Geometry

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  1. MTH 104 Calculus and Analytical Geometry Lecture # 8

  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. 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 Examples

  6. Techniques of differentiation

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

  8. Techniques of differentiation

  9. Techniques of differentiation 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 Derivative of numerator solution 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 derivative 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 of into (1) L.H.S=R.H.S

  25. Example Given that show that

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