3.8 Implicit Differentiation

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Photo by Vickie Kelly, 2003

Notes about the chain rule

?)( want weif what So,

1!by tion multiplica show tohavet don'just we;every time rulechain theusing are We

) wrt of derivative (the)(2,)(

2

12

ydxd

xxdxdxx

dxdythenxyIf

2 2 1x y

This is not a function, but it would still be nice to be able to find the slope.

2 2 1d d dx ydx dx dx

Do the same thing to both sides.

2 2 0dyx ydx

Note use of chain rule.

2 2dyy xdx

22

dy xdx y

dy xdx y

Example 1

22 siny x y

22 sind d dy x ydx dx dx

This can’t be solved for y.

2 2 cosdy dyx ydx dx

2 cos 2dy dyy xdx dx

22 cosdy xydx

22 cos

dy xdx y

This technique is called implicit differentiation.

1 Differentiate both sides w.r.t. x.

2 Solve for .dydx

Example 2

Example 3

3832for Find 223 xyyxxdxdy

Find the point(s) at which the curve

has horizontal or vertical

tangent lines.

Note product rule.

04484 22 yxyx

Example 4

Find if .2

2

d ydx

3 22 3 7x y

3 22 3 7x y

26 6 0x y y

26 6y y x

266xyy

2xyy

2

2

2y x x yyy

2

2

2x xy yy y

2 2

2

2x xyy

xyy

4

3

2x xyy y

Substitute back into the equation.

y

Example 5