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# IMPLICIT DIFFERENTIATION (2

By Ricardo Ellis,2014-01-07 06:39
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IMPLICIT DIFFERENTIATION (2

IMPLICIT DIFFERENTIATION (3.7)

Explicit vs. Implicit??? Meaning in everyday language??

Equations:

2 Explicit Relationship: y = x + 3x 7

23 Implicit Relationships: 2x y + 4x = 2

3 2 2x+ xy + y = 9-x

In the first of these two implicit relationships, it’s easy to solve for y

explicitly in terms of x. This would turn an implicitly defined function into an explicitly defined one. Do this now.

For implicitly defined functions that can’t be rewritten explicitly, we can find the derivative by using a special case of the chain rule. Recall that:

nn-1 D[f(x)]=n[f(x)][f’(x)] x

To find the derivative, y’ of an implicitly defined function, assume that we can express y explicitly as some function f(x).

23For example: 2x y + 4x = 2

23 Becomes: 2x [f(x)] + 4x = 2

Then we differentiate both sides of this equation with respect to x:

2 4x 3[f(x)]f’(x) + 4 = 0

2 4x 3[f(x)]f’(x) + 4 = 0

Then we solve this equation for f ’(x). Do this now.

This process is somewhat easier if we realize that since y=f(x),

nn D[f(x)] = D[y] xx

n-1 = n yy’

3 2example: 2x+ xy + y = 9-x

a. Find y’

b. Find the equation of the tangent line to the graph of this function at

the point (1,2).

2 example: Find y’ and y’’ if x= siny