Section 3.5 Implicit Differentiation

1

Example

If f(x) = (x7 + 3x5 – 2x2)10, determine f ’(x).

Now write the answer above only in terms of y if y = x7 + 3x5 – 2x2.

Answer: f΄(x) =10(x7 + 3x5 – 2x2)9(7x6 + 15x4 – 4x)

Answer: f ΄(x) = 10y9y΄

10

2 3

3 y

dy

dxd

edxd

x ydx

Examples

If y is some unknown function of x, find

Purpose

9x + x2 – 2y = 5 5x – 3xy + y2 = 2y

Easy to solve for y and differentiate

Not easy to solve for y and differentiate

Process wise, simply take the derivative of each side of the equation with respect to x and when we encounter terms containing y, we use the chain rule.

In equations like 5x – 3xy + y2 = 2y, we simply assume that y = f(x), or some function of x which is not easy to find.

Example

y3 = 2x

23 ' 2y y

Solving for y’, we have the derivative

2

2'

3y

y

3 2 22 3 ' 0xy x y y

Example

x2y3 = -7

Solving for y’, we have2 2 33 ' 2x y y xy

3

2 2

2 2'

3 3

xy yy

x y x

• Differentiate both sides of the equation.Since y is a function of x, every time we differentiate a term containing y, we need to multiply it by y’ or dy/dx.

• Solve for y’.• Every term containing y’ should be moved to the left by

adding or subtracting terms only.• Every term containing no y’ should be moved to the right

hand side.• Factor out y’ and divide both sides by the expression

inside ( ).

Implicit Differentiation

231. 3 2 5x y y

ExamplesDetermine dy/dx for the following.

2 2 22. 3 2 5x x y y

3 23. sin 2y x y

Find the equation of tangent line to the curve.

2 2 100 ; (8, 6)x y

3 2 32 8 19 ; 2y x y y x x

Example

arccosy xcos y x

sin 1y y

1

siny

y

2

1

1y

x

Find the derivative for

Derivative of Trig functions

21

1][arccos

xx

dx

d

21

1][arcsin

xx

dx

d

21

1][arctan

xx

dx

d

21

1]cotarc[

xx

dx

d

1||

1]secarc[

2

xxx

dx

d

1||

1]cscarc[

2

xxx

dx

d

Examples

Find the derivative for each function.

arccos(tan 3 )y x

2arcsec( )y x

5arctan 4arccoty x x

Examples

Find and simplify dy/dx for each function.

2arccos 1y x x x

2arcsin 1y x x

2arccot

1

xy x

x

2 arctan(5 )y x x

8arcsin( ) 8arccos( )y x x