HIGH-ORDER-DERIhrhrgddfVATIVES-IMPLICIT.pptx

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1. HIGHER ORDER DERIVATIVES AND IMPLICIT DIFFERENTIATION

2. HIGHER DERIVATIVES The derivative of a function f is itself a function and hence may have a derivative of its own. If is differentiable, then its derivative is denoted by and is called the second derivative of . As long as we have differentiability, we can continue the process of differentiating to obtain the third, fourth, fifth, and even higher derivatives of . These successive derivatives are denoted by  

3. Other common notations for higher derivatives are the following: first derivative: second derivative: n th derivative: The symbols are called Leibniz notations .

4. EXAMPLE 1 Find all the derivatives of the function

5. EXAMPLE 2 Find the second derivative of when x = 1.

6. EXAMPLE 3 Find  

7. IMPLICIT DIFFERENTIATION FUNCTIONS DEFINED EXPLICITLY AND IMPLICITLY There are two ways to define functions, implicitly and explicitly. Most of the equations we have dealt with have been explicit equations, such as  , so that we can write  where  . But the equation describes the same function. This second equation is an implicit definition of   as a function of  . As there is no real distinction between the appearance of   or   in the second form, this equation is also an implicit definition of   as a function of  .  

8. IMPLICIT DIFFERENTIATION An implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable. To give a function explicitly is to provide a prescription for determining the output value of the function   in terms of the input value  . By contrast, the function is implicit if the value of y is obtained from   by solving an equation of the form: .    

9. STEPS IN IMPLICIT DIFFERENTIATION 1. Differentiate both sides of the equation with respect to x. 2. Collect all the terms with on one side of the equation.   3. Factor out . 4. Solve for .

10. EXAMPLE 1

11. EXAMPLE