Implicit DifferentiationSection 3.7a

Consider the equation:3 3 9 0x y xy

Is this a function?

Is the equationdifferentiable?

If so, how do wedifferentiate?

1y f x

2y f x

3y f x

3 3 9 0x y xy

We use implicit differentiation, so namedb/c the functions are defined implicitly (hidden)within the original equation

1y f x

2y f x

3y f x

Implicit Differentiation Process We treat y as a differentiable function of x!!!

1.Differentiate both sides of the equation withrespect to x.

2. Collect the terms with dy/dx on one side ofthe equation.

3. Factor out dy/dx.

4. Solve for dy/dx.

Because of the Chain Rule, any time differentiatingany term containing a y, also multiply by dy/dx!!!

Initial Guided Practice

Find dy/dx: 2y x

2 1dyydx

1

2

dy

dx y

Does this answer make sense graphically?

Implicit Differentiation:

Solve for the derivative:

Initial Guided Practice

Find the slope of the circle at2 2 25x y

2 2 0dy

x ydx

First, find the slope of any point on the circle viaimplicit differentiation:

dy x

dx y

3 3

4 4

dy

dx

Slope at the given point:

Again, verify thisanswer graphically!

3, 4

2 2dyy xdx

Initial Guided PracticeShow that the slope dy/dx is defined at every point onthe graph of 22 siny x y

2 2 cosdy dy

x ydx dx

Imp Diff:

2 cos 2dy dy

y xdx dx

2 cos 2dy

y xdx

2

2 cos

dy x

dx y

This formula for dy/dxis defined at every point(x,y) except for thosepoints at which cos(y)=2…

Which never happens!!!

We can use all of this new info to expand the POWERof the POWER RULE!!! A proof:

First, let p and q be integers with q > 0 and suppose that:

q py x

1

1

p

q

dy px

dx qy

1 1q pdyqy px

dx

1

1

p

qp q

p x

q x

pq p qy x x 1p

p p q

p x

q x

1p p p qpx

q

1p qpx

q

Imp. Diff.!

Subst. for y

Law of Exp.

Final Answer!

The EXPANDED Power Rule:If n is any rational number, then

(If n < 1, then the derivative does not exist at x = 0)

WHY NOT???

Note: Before, the power rule worked for integers only… now, it works for any rational number power (i.e., fraction)

What is the derivative of thesquare root function???

1n ndx nx

dx

Additional Guided PracticeFor each of the following, find dy/dx.

2 31 6y x

1 34 1 6x

1 321 6 6

3

dyx

dx

Additional Guided PracticeFor each of the following, find dy/dx.

3 3 18x y xy

2 23 3 18 1dy dy

x y x ydx dx

2 23 18 18 3dy dy

y x y xdx dx

2 23 18 18 3dy

y x y xdx

Additional Guided PracticeFor each of the following, find dy/dx.

3 3 18x y xy

2

2

6

6

y x

y x

2 23 18 18 3dy

y x y xdx

2

2

18 3

3 18

dy y x

dx y x

Additional Guided PracticeFor each of the following, find dy/dx.

sinx y xy

1 cos 1dy dyy x ydx dx

cos 1dy dyy x ydx dx

cos 1dy

y x ydx

Additional Guided PracticeFor each of the following, find dy/dx.

sinx y xy

cos 1dy

y x ydx

1

cos

dy y

dx y x

1

cos

y

x y