Topics
11. Differentiation
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Standard Level

11.16 Differentiating logarithmic functions

Lesson
Worksheet
Practice
Worksheet
Natural logarithmic functions
1

Differentiate the following functions:

a
y = \ln x
b
y = 7 \ln x
c
y = \ln 6 x
d
y = 2 \ln 3 x
e
y = \ln 4 x - \ln 3
f
y = 4 \ln 5 x - x
g
y = x^{4} + 6 \ln 7 x
h
y = 4 \ln x - \dfrac{1}{x}
i
y = 3 \ln x + 5 \ln 2 x
j
y = \ln \left( 7 x + 3\right)
k
y = \ln \left(x^{4} + 2\right)
l
y = \ln \left(x^{2} + 7 x + 5\right)
m
y = \ln \left(x^{2} - 6 x + 9\right)
n
y = \ln \left(2 + 4 x + x^{3}\right)
o
y = \ln \left(\left( 4 x^{3} + 8 x^{2} - 9\right)^{3}\right)
p
y = \ln \left(\dfrac{x}{7}\right)
q
y = 4 \ln \left(\dfrac{x}{5}\right)
r
y = \ln \left(\dfrac{3}{x}\right)
s
y = \ln \left(8 - x\right) - 8
t
y = \ln \left(t^{2} - 5 t\right)
u
y = \ln \left(\dfrac{8}{x + 4}\right)
v
y = \ln \left(\dfrac{4}{3 x + 4}\right)
2

Consider the function y = \ln \left(\dfrac{x - 3}{x + 3}\right).

a

Let u = \dfrac{x - 3}{x + 3}. Rewrite y in terms of u.

b

Find \dfrac{d u}{d x}.

c

Find \dfrac{d y}{d u} in terms of x.

d

Find \dfrac{d y}{d x}.

3

Consider the function y = \ln x^{2}, where x \gt 0.

a

Rewrite the function without powers.

b

Hence determine y \rq \left( x \right).

c

Find the value of x at which y \rq \left( x \right) = \dfrac{1}{4}.

4

Consider the function y = \ln \left( 5 x - 2\right)^{4}.

a

Rewrite the function without powers.

b

Hence determine y \rq \left( x \right).

c

Find the exact value of x at which y' = \dfrac{1}{3}.

5

Differentiate the following functions:

a
y = \ln \sqrt{x}
b
y = \ln \sqrt{ 7 x}
c
y = \ln \sqrt{2 - 5 x}
6

Consider the function f \left( x \right) = \ln \left(\sqrt{x^{2} + 1}\right).

a

Find f \rq \left( x \right).

b

Find f \rq \left( 2 \right).

7

Consider the function f \left( x \right) = 4 \ln \left( 4 x^{2} + 3\right).

a

Find f \rq \left( x \right).

b

Find x, such that f \rq \left( x \right) = 4.

Multiple differentiation rules
8

Differentiate the following functions:

a
y = x \ln x
b
y = \left(\ln x\right)^{6}
c
y = \dfrac{\ln x}{x}
d
y = \dfrac{x}{\ln x}
e
y = x^{3} \ln x^{3}
f
y = 6 x^{4} \ln x
g
y = x^{5} \ln \left(x - 5\right)
h
y = \left(x + 6\right) \ln \left(x + 6\right)
i
y = \left(x^{2} + 6\right) \ln 2 x
j
y = e^{x} \ln x
k
y = \ln e^{ 8 x}
l
y = e^{\ln x}
m
y = e^{ 6 x} \ln 4 x
n
y = \ln \left(\log_{e} 4 x\right)
o
y = \dfrac{\ln x}{e^{x}}
p
y = \dfrac{\ln 3 x}{5 x^{4} + 2}
9

Consider the curve y = x^{3} \ln x.

a

Find the gradient function \dfrac{d y}{d x}.

b

Find the exact value of the gradient at the point where x = e^{4}.

10

If f \left( 6 \right) = 2 and f \rq \left( 6 \right) = 8, find the value of \dfrac{d}{dx} \left(\ln \left(f \left( x \right)\right)\right) at x = 6.

11

Given that f \left( x \right) = \ln \left(g \left( x \right)\right), g \left( 2 \right) = 4 and g \rq \left( 2 \right) = 9, evaluate f \rq \left( 2 \right).

12

Consider the function f \left( x \right) = x e^{ 3 x}.

a

Show that x e^{ 3 x} = e^{ 3 x + \ln x}.

b

Hence, find f \rq \left( x \right), without using the product rule.

13

Suppose that g \left( x \right) = \dfrac{\ln x}{f \left( x \right)}, for some function f \left( x \right).

a

Find an expression for g' \left( x \right) in terms of f \left( x \right) and its derivative f \rq \left( x \right).

b

If f \left( e^{3} \right) = 4 e^{3} and f \rq \left( e^{3} \right) = 2, find the value of g \rq \left( x \right) when x = e^{3}.

14

Consider the functions f \left( x \right) = k \ln x and g \left( x \right) = \ln k x, where k \gt 1 is a constant.

a

Find f \rq \left( x \right).

b

Find g \rq \left( x \right).

c

State how many times faster f \left( x \right) is increasing compared to g \left( x \right).

15

Given the following expressions:

  • \ln x = x - 1 - \dfrac{\left(x - 1\right)^{2}}{2} + \dfrac{\left(x - 1\right)^{3}}{3} - \dfrac{\left(x - 1\right)^{4}}{4} + ...
  • \dfrac{1}{x} = 1 - \left(x - 1\right) + \left(x - 1\right)^{2} - \left(x - 1\right)^{3} + \left(x - 1\right)^{4} - ...

Prove that \dfrac{d}{dx} \left(\ln x\right) = \dfrac{1}{x}.

Bases other than e
16

Consider the function y = \log_3 x.

a

Make x the subject.

b

Rewrite the expression for x with base e.

c

Find \dfrac{dx}{dy}.

d

Write \dfrac{dx}{dy} in terms of x.

e

Hence find \dfrac{dy}{dx}.

17

Differentiate the following functions:

a
f \left( x \right) = \log_{2} x
b
f \left( x \right) = \log_{2} 4 x
c
f \left( x \right) = \log_{3} \left( 2 x - 4\right)
d
f \left( x \right) = \log_{10} \left(1 - x\right)
e
f \left( x \right) = \log_{5} \left( 4 x + 2\right) - 6 x
f
f \left( x \right) = \log_{6} \left(2 - 7 x\right)
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