# Derivative of the inverse of a function

Textbook pages 223-228

In some situations, a function is defined as the inverse of another function:

Examples:

•

•

•

•

When this is the case, to calculate the derivative those functions we use the
inverse rule.

Rule:

Examples:

• Let f(x) = e^{x}, what is the derivative of ln(x) = f^{-1}(x) ?

• Let f(x) = arcsin(x), what is f'(x)?

• What is the derivative of arctan(x)?

Proof:

**4.4.19 Derivatives of logarithmic functions**

Textbook pages 229-231

A logarithmic function in base a, f(x) = log_{a}(x), is defined as the inverse of
the expo-

nential function in base a. So, using the inverse rule we find that

**Examples:**

• The derivative of the natural logarithm is

• The derivative of log_{2}(x) is

Chain rule for logarithms: Given a function f(x), what is the derivative of ln[f(x)]?

Examples:

• f(x) = ln(3x + 1):

• f(x) = ln(sin(x)):

**4.4.20 Logarithmic differentiation**

Textbook pages 231-233

In the previous examples, we saw that taking the derivative of the logarithm
of a com-

plicated function involving products and powers is actually very easy. By
comparison, taking

the derivative of the function itself could be more complicated. Based upon this
idea, we now

introduce the concept of logarithmic differfentiation. The following example
will illustrate

how powerful this method is.

Example: Find the derivative of

Idea:

• Take the logarithm of both sides of this equality, and expand the log.

• Take the derivative of both sides of this equality

• Multiply both sides by f(x) to obtain f'(x)

Example: The logarithmic differentiation is also useful to find derivatives
of complicated

functions such as f(x) = x^{x}.

Proof of the power rule in the general case: Earlier in the lecture, we
proved the

power rule

for positive integer vales of α. We are now able to
prove it in the general case, using

logarithmic differentiation.

**• Inverse Rule:**

Using the inverse rule, find the derivative of f(x) = arccos(x).

**• Derivatives of logarithmic functions**

Textbook problems page 234 number 23, 27, 31, 33, 39, 43, 49

**• Logarithmic differentiation:
**– Let f(x) = g(x)

^{α}. Using the chain rule, prove that Prove

the same result using logarithmic differentiation (i.e. say ln(f(x)) = ln(g(x)

^{α})

and take the derivative on both sides).

– Textbook problems page 234 number 63, 65, 67, 71, 73, 75.

**• Miscellaneous derivatives problems**

Textbook review problems page 243 number 1,3,5,7,15,23,30,36