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Chapter 2 Derivatives (Part 2) Outline 1. Implicit Differentiation

2. Derivative of Inverse Functions

3. Inverse Trigonometric Functions

4. Linear Approximation

5. Differentials

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Implicit Differentiation

If a quantity can be expressed in terms of a variable (or quantity) as

we say that is defined explicitly as a function of .

There are, however, many important circumstances where and are related by an equation

such that cannot be solved as a single explicit function of as above.

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EX Suppose two quantities are related by the equation

If we try to solve in terms of , then

We get two functions and for . Plugging or for , then

is true.

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EX The equation

gives the so-called the folium of Descartes: whose graph is as shown below.

The second figure suggests that there are 3 possible functions for that satisfy the equation. However, the formula may not be easy to write down!

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Def We say is an implicit function induced from if

is true. We also say is implicitly defined from .

We say is an implicit function induced from if

is true. We also say is implicitly defined from .

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EX and are implicitly defined from the equation

The functions whose graphs are displayed below are implicit functions induced from the equation

Remark For many circumstances, one may not be able to write down formulas of implicit functions!

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Implicit differentiation

Let be an implicit function induced from the equation

To find we perform the implicit differentiation:

(1) Keep in mind that .

(2) Diff w.r.t to get

Don’t forget to use the chain rule.

(3) Solve appeared in (2).

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EX Find for a function defined implicitly by

Also calculate at the point .

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EX Find if

and determine the tangent line at the point .

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EX Find if

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Inverse Functions

Given a function , a function is said to be the inverse function of if

is often denoted by .

One can find the inverse function by

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EX The function

is the inverse function of

because

has no inverse because it is

not one-to-one.

?

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EX (1) Find the inverse function of

(2) Find the inverse function of

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Derivative of Inverse Functions Assume is a one-to-one differentiable function with the inverse . If

then is differentiable at and

Thus at any where ,

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Proof We use implicit differentiation.

Since , this means is implicitly defined by the equation

By implicit diff and the chain rule, we have

So we obtain

hence

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EX If , find .

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EX Find the formula for the inverse of the function and find the derivative .

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Inverse Trigonometric Functions

with Dom is not 1-1.

It is 1-1 if we consider

Def (The inverse of sine)

Dom and Rng

.

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with Dom is not 1-1.

It is 1-1 if we consider

Def (The inverse of cosine)

Dom and Rng .

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Derivative Formulas

Proof We show

using

implicit diff. That means

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Def (Other inverse trig functions)

Dom , Rng

.

Dom , Rng

.

Dom and Rng .

Dom and Rng

.

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Derivative formulas

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EX (1) Find the value of

(2) Find the limit

Remark The graph of

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EX Find the derivative

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Linear Approximation

Let be a function which is differentiable at . Then the following limit exists:

This means

as . Thus when is small, we have the approximation

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Def The approximation

is called the linear approximation. The function

is called the linearization of at .

Remark For many functions, calculating is difficult whereas

can be easily computed. The error

of this approx. gets smaller as .

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EX Find the linearization of

at .

Then use it to approximate and

.

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Differentials

For a function , is called an independent variable (i.e. it can take any value freely) and is called a dependent variable (i.e. its value depends on ). Def (Differentials)

Let be a differentiable function.

The differential of is an independent variable (different from ) denoted by

The differential of is a dependent variable (different from ) denoted by

and it is related to and by

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Using the independent variable , the linearization at is

By linear approximation,

So is used to approximate the change (or error) in given a change (or error) in :

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EX Find the differential where

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EX The circumference of a sphere was measured to be cm with a possible error of cm.

(1) Use the differential to approximate the maximum error in the calculated area of the sphere.

(2) What is the relative error? What is the percentage error?

Remark