1. Derivatives of Logarithmic and Exponential functions

2. One-to-one functions
Definition: A function f is called a one-to-one function if it never takes on the same value twice; that is
f(x1) ≠ f(x2) whenever x1 ≠ x2.
Horizontal line test: A function f is one-to-one if and only if no horizontal line intersects its graph more than once.
Examples: f(x) = x3 is one-to-one
but f(x) = x2 is not.

3. Inverse functions
Definition: Let f be a one-to-one function with domain A and range B. Then the inverse function f -1 has domain B and range A and is defined by
for any y in B.
Note: f -1(x) does not mean 1 / f(x) .
Example: The inverse of f(x) = x3 is f -1(x)=x1/3
Cancellation equations:

4. How to find the inverse function of a one-to-one function f
Step 1: Write y=f(x)
Step 2: Solve this equation for x in terms of y (if possible)
Step 3: To express f -1 as a function of x, interchange x and y. The resulting equation is y = f -1(x)
Example: Find the inverse of f(x) = 5 - x3

5. Another example:
Solve for x:
Inverse functions are reflections about y = x.
Switch x and y:

6. Derivative of inverse function
First consider an example:
Slopes are reciprocals.
At x = 2:
At x = 4:

7. Calculus of inverse functions
Theorem: If f is a one-to-one continuous function defined on an interval then its inverse function f -1 is also continuous.
Theorem: If f is a one-to-one differentiable function with inverse function f -1 and f ′ (f -1 (a)) ≠ 0, then the inverse function is differentiable and
Example: Find (f -1 )′ (1) for f(x) = x3 + x + 1
Solution: By inspection f(0)=1, thus f -1(1) = 0
Then

8. Logarithmic Functions
Consider where a>0 and a≠1
This is a one-to-one function, therefore it has an inverse.
The inverse is called the logarithmic function with base a.
Example:
The most commonly used bases for logs are 10:
and e:
is called the natural logarithm function.

9. Properties of Logarithms
Since logs and exponentiation are inverse functions, they “un-do” each other.
Product rule:
Quotient rule:
Power rule:
Change of base formula:

10. Derivatives of Logarithmic and Exponential functions
Examples on the board.

11. Logarithmic Differentiation
The calculation of derivatives of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms.
Step 1: Take natural logarithms of both sides of an equation y = f (x) and use the properties of logarithms to simplify.
Step 2: Differentiate implicitly with respect to x
Step 3: Solve the resulting equation for y′
Examples on the board