Calculus AB

Calculus AB

Chapter 4

More Derivatives

AP Classroom

2.7, 3.2, 3.3, 3.4

R7/8, R9, R10

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4.2 Implicit Differentiation

AP: 3.2

When you need to find the derivative of a function given implicitly, it is not necessary to

solve the equation for y in terms of x.

Instead, we use the method called implicit differentiation. It consists of 3 steps:

1) Differentiate both sides of the equation with respect to x. When you take the

derivative of a term that includes y, think of y as y(x) and apply the Chain Rule

dy

(multiply by

or , y? )

dx

2) Collect the terms with y? on one side of the equation and the terms without it on the

other side.

3) Solve for y?

Example 1

Differentiate x2 - xy + y2 = 21

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Example 2

Determine the slope of the tangent line to the circle x2 + y2 = 13 at the point (2,3)

Example 3

Write the equation of the tangent line to the curve x 2 cos 2 y ? sin y = 0 at the point (0, ? )

Read Example 5 on page 165

Assign p. 167

numbers 1 - 19 (odd`s only), 23, 33, 35

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4.3 Part 1 Derivatives of Inverse Functions

AP: 3.3

A function is said to be one-to-one if, for each output there is a single input.

y

y

10

10

5

5

- 10

-5

5

10

x

- 10

-5

5

-5

-5

- 10

- 10

one-to-one

10

x

not one-to-one

A function must be one-to-one to have an inverse that is a function.

Theorem:

Let

Then

be a one to one continuous function on an interval .

is strictly increasing or decreasing.

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