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Implicit Implicit DifferentiationDifferentiation
ObjectiveObjective
To find derivatives implicitly.To find derivatives implicitly.
Implicit DifferentiationImplicit Differentiation
Fails the vertical line test!
Implicit DifferentiationImplicit Differentiation
Let’s say we had a circleLet’s say we had a circle
dy xdx y
01 0dy
dx
I can see the slope at (0, -1) is 0, I can see the slope at (0, -1) is 0, and I know that a derivative and I know that a derivative
equation gives slope. But how can I equation gives slope. But how can I find the derivative of that equation?find the derivative of that equation?
Because once I had it I could Because once I had it I could find the slope of any point, find the slope of any point, even if it wasn’t obvious from even if it wasn’t obvious from the picture!the picture!
Implicit DifferentiationImplicit Differentiation
dy xdx y
12
12
1dydx
Implicit DifferentiationImplicit Differentiation
dy xdx y
1212
1dydx
Implicit DifferentiationImplicit Differentiation
dy xdx y
10 und.dy
dx
For some equations we could do it the old fashioned way, by isolating y.
2Find for 3 7 14dydx x y xy
2(3 7 ) 14y x x
2
14
3 7y
x x
2 114(3 7 )y x x
2 2
14(6 7)
(3 7 )dydx
x
x x
dydx 14 2(3 7 )x x 2 (6 7)x
Implicit DifferentiationImplicit Differentiation
So far in calculus, we have worked So far in calculus, we have worked only with equations in only with equations in explicit formexplicit form. .
An equation is in An equation is in explicit formexplicit form when when one variable is directly equal to an one variable is directly equal to an expression made up of the other expression made up of the other variable.variable.
2Examples: 3 5, 16 20y x p t t
Implicit DifferentiationImplicit Differentiation
Sometimes equations are not in Sometimes equations are not in explicit form, but in a more explicit form, but in a more complicated form in which it is difficult complicated form in which it is difficult or impossible to express one variable or impossible to express one variable in terms of the other. Such equations in terms of the other. Such equations are in are in implicit formimplicit form..
2 2 3 2Examples: 3 7 14, 3 2 2x y xy x y xy
Implicit DifferentiationImplicit Differentiation
Implicit differentiation uses the Implicit differentiation uses the chain rule in a creative way to chain rule in a creative way to find the derivative of equations in find the derivative of equations in implicit form.implicit form.
derivativeuuuuuuuur
Implicit DifferentiationImplicit Differentiation
dxdx 6x
2 3 2Find for 3 2 2dydx x y xy
23xderivativeuuuuuuuur 6x
2y
32y 26y dydx
2xy xderivativeuuuuuuuur dy
dx2y (1) 22 dy
dxxy y
2 0derivativeuuuuuuuur
dxdx
2 26 6 2 0dy dydx dxx y xy y
2 26 2 6dy dydx dxy xy x y
2 2(6 2 ) 6dydx y xy x y
2
2
6
6 2dydx
x y
y xy
2(6 )
2 (3 )
x y
y y x
2 3 2Find for 3 2 2dydx x y xy
Implicit DifferentiationImplicit Differentiation2 3 3Find for 6 3dy
dx y x y y
2 dydxy
23x 23 dydxy 3 dy
dx
22 3 3dy dy dydx dx dxy y 23x
2 2(2 3 3) 3dydx y y x
2
2
3
2 3 3dydx
x
y y
Implicit DifferentiationImplicit Differentiation3 2 4Find for 10 5dy
dx x x y y
23x 2x dydx 0
2 340dy dydx dxx y 23 2x xy
y (2 )x 340 dydxy
2 3 2( 40 ) 3 2dydx x y x xy
2 3
(3 2 )
40dydx
x x y
x y
Implicit DifferentiationImplicit Differentiation3 3Find at (2, 1) for 7dy
dx y y x 23 dydxy 23x
2(3 7)dydx y 23x
7 dydx
2
2
3
3 7dydx
x
y
2
2
3(2)
3(1) 7dydx
12
10
6
5
Implicit DifferentiationImplicit Differentiation2 2Find the slope of =100 at (6, 8)x y
2x 0
2 dydxy 2x
2 dydxy
2
2dydx
x
y
6
8dydx
3
4
x
y
Implicit DifferentiationImplicit Differentiation3 2 2Find at (1, 3) for 5 4dy
dx y y y x 23 dydxy 5 dy
dx
2(3 2 5) 2dydx y y x
2 dydxy
2
2
3 2 5dydx
x
y y
2x 0
23 2 5dy dy dydx dx dxy y 2x
Implicit DifferentiationImplicit Differentiation
2
2
2(1)
3( 3) 2( 3) 5dydx
2
16
1
8
2
2
3 2 5dydx
x
y y
Implicit DifferentiationImplicit Differentiation
Find the equation of the tangent line to the Find the equation of the tangent line to the graph of:graph of: 3 3 6 at (3, 3)x y xy
23x 23 dydxy 6x dy
dx y (6)
23 6dy dydx dxy x 26 3y x
2 2(3 6 ) 6 3dydx y x y x
2
2
3(2 )
3( 2 )dydx
y x
y x
Implicit DifferentiationImplicit Differentiation2
2
2
2dydx
y x
y x
2
2
2(3) (3)
(3) 2(3)dydx
3
3dydx
1
y mx b
3 1(3) b
3 3 b
6 b
6y x
ConclusionConclusion
An equation is in An equation is in explicit formexplicit form when one variable when one variable is directly equal to an expression made up of the is directly equal to an expression made up of the other variable. other variable.
An equation is in An equation is in implicit formimplicit form when neither when neither variable is isolated on one side of the equal sign.variable is isolated on one side of the equal sign.
Find the derivative of a relation by differentiating Find the derivative of a relation by differentiating each side of its equation implicitly and solving for each side of its equation implicitly and solving for the derivative as an unknown. the derivative as an unknown.
