Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Pre-calculus Concepts:
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Find the inverse of f ( x ) = −1 − x .
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Calculus of Inverse Functions
f ( x ) = −1 − x
Find
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
( f )′ ( 4 ) .
−1
1. Find the x − value that gives an output of 4
2. Plug this x − value into the derivative of f
3. Your answer is the reciprocal of this value
With this function we could have found the inverse function first and then taken its derivative.
We won't always be able to do this.
f ( x ) = 2 x + 4sin x + 2 cos x
3
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
′
Find ( f −1 ) ( 2 ) .
1. Find the x − value that gives an output of 2
2. Plug this x − value into the derivative of f
3. Your answer is the reciprocal of this value
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
f ( x) = ax
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
e ≈ 2.7128...
f ( x ) = ex
The natural exponential function
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
The Chain Rule with the exponential function
Let y = xe − x
a ) Find the x − values so that y ′ ( x ) = 0
b ) Find the x − values so that y ′′ ( x ) = 0
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
f ( x ) = ln ( x )
The natural logarithm function
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
The derivative of the logarithm function
The Chain Rule with the logarithm function
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
Logarithmic Differentiation
Purpose: To be able to take the derivative of functions like y =
y=
( x)
Step 1 :
Step 2 :
Step 3 :
x
Math 103 – Rimmer
3.8 Inverse Functions
and Log. Functions
( x)
x