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2.5 Implicit Differentiation

Niagara Falls, NY & Canada

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

2 2 1x y Write y explicitly as a function of x and take the derivative.

Calculus Warm-up before 2.5

Distinguish between functions written in implicit form and explicit form.

Use implicit differentiation to find the derivative of a function.

Objectives

Implicit and Explicit Functions

Implicit and Explicit Functions

Most functions have been expressed in explicit form.

For example, in the equation

the variable y is explicitly written as a function of x.

Some functions, however, are only implied by an equation. For instance, the function y = 1/x is defined implicitly by the equation xy = 1.

Explicit form

Suppose you were asked to find dy/dx for this equation. You could begin by writing y explicitly as a function of x and then differentiating.

This strategy works whenever you can solve for y and write it explicitly as a function of x.

You cannot, however, use this procedure when you are unable to solve for y as a function of x.

Implicit and Explicit Functions

For instance, how would you find dy/dx for the

equation

where it is very difficult to express y as a

function of x explicitly? To do this, you can use

implicit differentiation.

Implicit and Explicit Functions

Use implicit differentiation when it is impossible or impractical to express y as a function of x explicitly.

To find dy/dx implicitly, realize that differentiation is taking place with respect to x.

This means that when you differentiate terms involving x alone, you can differentiate as usual. However, when you differentiate terms involving y, you must apply the Chain Rule, because you are assuming that y is defined implicitly as a differentiable function of x.

Example 1 – Differentiating with Respect to x

Example 1 – Differentiating with Respect to x, cont’d

2dxy

dx 2 2

d dyx y y x

dx dx

2 2dy

y xydx

2 2 1x y

Its not necessary to solve for y because we can take this derivative implicitly.

2 2 1d d d

x ydx dx dx

Do the same thing to both sides.

2 2 0dy

x ydx

Note use of chain rule.

2 2dy

y xdx

2

2

dy x

dx y

dy x

dx y

22 siny x y

22 sind d d

y x ydx dx dx

This can’t be solved for y.

2 2 cosdy dy

x ydx dx

2 cos 2dy dy

y xdx dx

22 cosdy

xydx

2

2 cos

dy x

dx y

This technique is called implicit differentiation.

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

2 Solve for .dy

dx

Find .dy

dx

You try:

We need the slope. Since we can’t solve for y, we use

implicit differentiation to solve for .dy

dx

Find the equations of the lines tangent and normal to the

curve at .2 2 7x xy y ( 1, 2)

2 2 7x xy y

2 2 0dydy

x yx ydxdx

Note product rule.

2 2 0dy dy

x x y ydx dx

22dy

y xy xdx

2

2

dy y x

dx y x

2 2 1

2 2 1m

2 2

4 1

4

5

Find the equations of the lines tangent and normal to the

curve at .2 2 7x xy y ( 1, 2)

4

5m tangent:

42 1

5y x

4 42

5 5y x

4 14

5 5y x

normal:

52 1

4y x

5 52

4 4y x

5 3

4 4y x

Higher Order Derivatives

Find if2

2

d y

dx3 22 3 7x y

3 22 3 7x y

26 6 0x y y

26 6y y x

26

6

xy

y

2xy

y

2

2

2y x x yy

y

2

2

2x xy y

y y

2 2

2

2x xy

y

x

yy

4

3

2x xy

y y

Substitute back into the equation.

y

Derivative formulas include the chain rule!

1n nd duu nu

dx dx sin cos

d duu u

dx dx

cos sind du

u udx dx

2tan secd du

u udx dx

etcetera…

If formulas on a memorization sheet are written with

instead of . Don’t forget to include the term!

uudu

dx

Homework

2.5 Day 1: pg.146: 1-15 odd, 21-31 odd, 45, 57

Day 2: MMM 53-54