3.8 Implicit Differentiation Niagara Falls, NY & Canada Photo by Vickie Kelly, 2003.
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Transcript of 3.8 Implicit Differentiation Niagara Falls, NY & Canada Photo by Vickie Kelly, 2003.
3.8 Implicit Differentiation
Niagara Falls, NY & Canada
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