L23 - 1
Lecture 23: Section 4.5
Derivatives of Logarithmic Functions
dy
Let y = loga(x). Can we find an expression for ?
dx
d
We have (loga(x)) =
dx
d
(ln(x)) =
dx
L23 - 2
ex. Find f 0(e) for f (x) = x ln x.
d
ex. Find (log2 x)3.
dx
L23 - 3
The Chain Rule for Logarithmic Functions
Let f be a differentiable function of x.
Then for f (x) > 0,
d
[ln(f (x))] =
dx
or if u is a differentiable function of x,
d
(ln(u)) =
dx
L23 - 4
ex. Find
d
(log3(e3t − 1)).
dt
Be careful of domain!
ex. Find the equation of each horizontal tangent
line of f (x) = ln(3x − x3).
L23 - 5
ex. Find f 0(x) if f (x) = ln |x|.
6
-
?
ex. Find each x-value at which f (x) = ln |3x − x3|
has a horizontal tangent line.
L23 - 6
ex. Write the equation of the tangent line of
f (x) = ln(ln x) at x = e.
L23 - 7
ex. Find
√
d
(ln x).
dx
Recall the following: If x > 0 and y > 0,
1. ln(xy) =
x
2. ln
=
y
3. ln(xy ) =
L23 - 8
We can use these properties to write a complicated
logarithmic function into a form involving sums and
differences, which are easier to differentiate.
r
x2 + 2x
0
.
ex. Find f (x) if f (x) = ln
2x − 6
L23 - 9
Additional Topic
We can use the process of Logarithmic Differentiation to find the derivative of a complicated expression which does not contain logarithms initially:
1.
2.
3.
L23 - 10
ex. Find the slope of the tangent line to
√
3 3
x − 3x − 8
f (x) =
at x = 0.
2x
e (x + 1)
L23 - 11
ex. Find f 0(x) for f (x) = xx.
L23 - 12
Now You Try It!
1. Find the slope of the tangent line to y = log3 (2x2 + 4x) at x = 1.
2. Find
d
[ln(ln(x2 + 6x))].
dx
3. Find the slope of the tangent line to the graph of
xey + 2 = 2x − ln(y + 1) at the point (2, 0).
√
4. Find each horizontal tangent line to the graph of f (x) = ln
r
0
5. Find f (ln 3) if f (x) = ln
4x − 3
.
(x + 1)3
ex + 1
.
ex − 1
6. Suppose that $1000 is deposited into an account with annual interest rate r% compounded annually.
(a) Find a formula for A, the amount of money in the account after
t years (see lecture 7). Express the interest rate as a fraction
r
.
100
(b) Find an expression for T , the amount of time it takes for the
money in the account to double.
dT
. Then find the derivative for r = 4.
dr
Include units in your answer.
(c) Find
√
2
dy
ex −2 6 + 3x
7. Find
if y =
. Find the slope of the tangent line at
dx
(3x − 1)4
x = 1.
√
8. Find the equation of the tangent line to f (x) = x
x
at x = 4.