Warm-Up: Find f’(x) if f(x)=(3x2-6x+2)3

SECTION 6.4: IMPLICIT DIFFERENTIATION

Objective: Students will be able to… Take the derivative of implicitly defined functions.

PUT YOUR PARTY HATS ON!!! ITS GOING TO BE FUN!

•Almost all the functions we have worked with so far have been of the form y = f(x)

In these cases, y is given explicitly in terms of x

Examples:

y = 2x + 5 , y = x2 + x + 6 , f(x)=(3x2-6x+2)3

5xy – 4x = 2This is an implicit function (not in form y =). However, it can easily be solved for y:

•Not all implicit functions can be rewritten explicitly

•Example:

y5 + 7y3 + 6x2y2 + 4yx3 + 2 = 0

In such cases, it is possible to find the derivative, dy/dx by a process called implicit differentiation.

Implicit Differentiation

•We assume y is a function defined in terms of x

•We differentiate using the chain rule:

Explicit: Implicit:

232332 23 xxdx

d dx

dyyy

dx

d 23 3

•Variables disagree•Derivative of inner function is implicitly defined

Implicit Differentiation

To find dy/dx for an equation containing x and y:1. Differentiate on both sides of the equation with respect to x, keeping in

mind that y is assumed to be a function of x. When differentiating x terms, take derivative as usual When differentiating y terms, you assume y is implicitly

defined as a function of x. Use chain rule.

2. Place all terms with dy/dx on one side of the equal sign, and all terms without dy/dx on the other side.

3. Factor out dy/dx, and then solve for dy/dx.

Examples:Find the derivative of the following functions.

1. y3 + y2 – 5y – x2 = -4

2. x2 – 2xy + y3 = 5

Find dy/dx.

1. sinx + x2y = 10 2. yx

yxx

3