Chapter 3.8: Derivatives of Inverse Functions and
Logarithms
3.7.
1
Inverse Function
Observation: Tangent lines will also flip
across the line y = x.
y = f (x)
y =x
y = f −1 (x)
4
If y = mx + b is a tangent line to
y = f (x) at (a, f (a)),
2
y
then x = my + b is the tangent line to
y = f −1 (x) at (f (a), a).
0
Note x = my + b becomes
y = m1 (x − b).
−2
−2
−1
0
1
2
3
4
5
x
The slope
1
m
reciprocal.
Functions are related to their inverses by
flipping across the line y = x.
Recall: f has to passes the horizontal line test to have inverse.
3.7.
2
Derivatives of Inverse Functions
1
d h −1 i
f (x) = 0 −1
dx
f (f (x))
Example: f (x) = x 3 + 4x + 5.
Compute f −1 (x) at x = 10.
Derivation can be done using implicit
function approach:
x =x
−1
f (f (x)) = x
i
d h i
d h −1
f (f (x)) =
x
dx
dx
h
i
d −1
f (x) = 1
f 0 (f −1 (x)) ·
dx
h
d −1 i
1
f (x) = 0 −1
dx
f (f (x))
3.7.
3
Recall Exponential and Log Functions
f (x) = e x ↔ f −1 (x) = ln(x)
y = ex
y =x
y = ln(x)
4
e a e b = e a+b ↔ ln(ab) = ln(a) + ln(b)
2
y
b
(e a ) = e ab ↔ ln ab = b ln(a)
a
ea
= e a−b ↔ ln
= ln a − ln b
b
e
b
0
−2
−2
−1
0
1
2
3
4
5
e ln x = x ↔ ln (e x ) = x
x
e 1 = e ↔ ln (e) = 1
a = e ln a
3.7.
loga x =
ln x
ln a
e 0 = 1 ↔ ln (1) = 0
4
Derivative of ln(x)
Using inverse derivative
Using implicit function differentiation
d h −1 i
1
f (x) = 0 −1
dx
f (f (x))
y = ln x
ey = x
d
d
(e y ) =
(x)
dx
dx
i
d h
ln(x) =
dx
i
d h
ln(x) =
dx
Interesting:
3.7.
d k
x = kx k−1 . How could you get x −1 on the right-hand side?
dx
5
Examples for Derivatives with ln
1.
d
[ln(3x)] =
dx
2.
d t[ln(t)]2 =
dt
d
3.
dx
ln(x)
=
x
4.
d x
[3 ] =
dx
5.
d x
[x ] =
dx
Logarithmic Differentiation
Idea: Instead of using
Example: y =
p
x(x + 1)
y = f (x)
dy
= f 0 (x)
dx
First take ln of both sides and then take
the derivative.
y = f (x)
ln(y ) = ln(f (x))
d
1 dy
=
[ln(f (x)]
y dx
dx
dy
d
=y·
[ln(f (x)])
dx
dx
d
dy
= f (x) ·
[ln(f (x)])
dx
dx
Sometimes ln(f (x)) is simpler for taking derivative than f (x).
3.7.
7
Logarithmic Differentiation Examples
Use logarithmic differentiation to compute the derivatives of following functions.
Note we computed these already with different method.
y = 3x
3.7.
y = xx
8
Logarithmic Differentiation Examples Round II
Use logarithmic differentiation to compute the derivatives of following:
r
t
f (t) = t(t + 1)(t + 2)
y= 3
t +1
3.7.
9
Chapter 3.8 Recap
i 1
d h
ln(x) =
dx
x
1
d h −1 i
f (x) = 0 −1
dx
f (f (x))
If y = f (x) then
i
d h
d
ln(y ) =
[ln(f (x)]
dx
dx
3.7.
10