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Implicit Differentiation. Objective: To find derivatives of functions that we cannot solve for y. Implicit Differentiation. It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation (but often it is easier to do so). - PowerPoint PPT Presentation
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Implicit Differentiation
Objective: To find derivatives of functions that we cannot solve for y.
Implicit Differentiation
• It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation (but often it is easier to do so).
• Find dy/dx for . Can we solve this for y?xyxy 1
Implicit Differentiation
• It is not necessary to solve an equation for y in terms of x in order to differentiate the function defined implicitly by the equation.
• For example, we can take the derivative of with the quotient rule:
1
1
x
xy
22 )1(
2
)1(
)1)(1()1)(1(
xx
xx
dx
dy
Implicit Differentiation
• We can also take the derivative of the given function without solving for y by using a technique called implicit differentiation. We will use all of our previous rules and state the independent variable.
xyxy 1
Example 2
• Use implicit differentiation to find dy/dx if22 sin5 xyy
Example 2
• Use implicit differentiation to find dy/dx if22 sin5 xyy
xdx
dyy
dx
dyy 2cos10
Example 2
• Use implicit differentiation to find dy/dx if22 sin5 xyy
xdx
dyy
dx
dyy 2cos10
xdx
dyyy 2)cos10(
Example 2
• Use implicit differentiation to find dy/dx if22 sin5 xyy
xdx
dyy
dx
dyy 2cos10
xdx
dyyy 2)cos10(
yy
x
dx
dy
cos10
2
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
048 dx
dyyx
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
048 dx
dyyx
y
x
dx
dy 2
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
y
x
dx
dy 2
22
2 2)2(
ydxdyxy
dx
yd
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
y
x
dx
dy 2
22
2 2)2(
ydxdyxy
dx
yd
22
2
222
yyx
xy
dx
yd
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
y
x
dx
dy 2
22
2 2)2(
ydxdyxy
dx
yd
22
2
222
yyx
xy
dx
yd
3
22
2
2 42
y
xy
dx
yd
Example 3
• Use implicit differentiation to find if 2
2
dx
yd .924 22 yx
y
x
dx
dy 2
22
2 2)2(
ydxdyxy
dx
yd
22
2
222
yyx
xy
dx
yd
3
22
2
2 42
y
xy
dx
yd
32
2 9
ydx
yd
Example 4
• Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).012 xy
Example 4
• Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).• We know that the slope of the tangent line means
the value of the derivative at the given points.
012 xy
Example 4• Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).• We know that the slope of the tangent line means
the value of the derivative at the given points.
012 xy
012 dx
dyy
ydx
dy
2
1
Example 4• Find the slopes of the tangent lines to the curve at the points (2, -1) and (2, 1).• We know that the slope of the tangent line means
the value of the derivative at the given points.
012 xy
012 dx
dyy
ydx
dy
2
1
2
1
12
yxdx
dy
2
1
12
yxdx
dy
Example 5
Use implicit differentiation to find dy/dx for the equation .xyyx 333
Homework
• Page 241-242• 1-23 odd• 27 (just use implicit)
