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6.07 Differentiation and logarithms

Worksheet
Differentiate y = ln x
1

Consider the graph of y = \ln x.

a

State whether the function is increasing or decreasing.

b

State whether the gradient to the curve is negative at any point on the curve.

c

Describe the change in the gradient of the tangent as x increases.

d

Describe the change in the gradient of the tangent as x gets closer and closer to 0.

1
2
3
4
5
6
7
8
9
10
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
2

Consider the function f \left( x \right) = \ln x.

a
Find f'(x).
b

Find f\rq \left(5\right).

c

State the value of x for which f\rq \left(x \right) is undefined.

3

Consider the function f \left( x \right) = \ln x.

a

Find the x-intercept.

b

State the equation of the tangent line to the curve at the point where it crosses the x-axis.

4

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}.

Differentiate y = ln(f(x))
5

Consider the function y = \ln a x, where a is a constant with a\gt 0 and x \gt 0.

a

Let u = a x. Rewrite the y in terms of u.

b

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

c

Hence find \dfrac{d y}{d x}.

d

Given f \left( x \right) = \ln 6 x. Find:

i
f\rq \left( x \right)
ii
f\rq \left( 3 \right)
6

Differentiate the following functions:

a

y = 7 \ln x

b

y = -2\ln x

c

y = \ln 6 x

d

y = \ln 4 x - \ln 3

e

y = \ln \left(\dfrac{x}{7}\right)

f

y = 2 \ln 3 x

g

y = 4 \ln \left(\dfrac{x}{5}\right)

h

y = \ln \left(8 - x\right) - 8

i

y = \ln \left(2 + 4 x + x^{3}\right)

j
y = \ln \left(x^{4} + 2\right)
k
y = \ln \left(t^{2} - 5 t\right)
l
y = \ln \left(x^{2} + 7 x + 5\right)
m

y = \ln \left( 4 x + 5\right)

n

y = \ln \left(3 - 2 x\right)

o

y = 2 \ln \left( 6 x - 5\right)

p

y = 3 \ln \left( \dfrac{1}{4} x - 6\right)

q

y = \ln \left( 7 x + 3\right)

r

y = \ln \left(\sin x\right)

7

Consider the functions f \left( x \right) = \ln x and g \left( x \right) = \ln 3 x.

a

Sketch both functions on the same Cartesian plane.

b

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

c

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

d

What can be concluded about the tangents of the curves at any given x-value?

8

Consider the function y = \ln \left( - x \right).

a

State the domain of this function.

b

Find \dfrac{dy}{dx}.

c

Hence, find the gradient of the tangent to the curve at x = - 5.

9

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}.

10

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

How many times faster is f \left( x \right) increasing than g \left( x \right)?

11

For each of the following curves:

i

Find the derivative of the function.

ii

Find the exact value of the gradient of the tangent at the given point.

a

y = \ln 2 x at the point where x=5.

b

y = 4 x + \ln 3 x at the point where x = \dfrac{1}{5}.

c

y = \ln \left(x^{2} + 5\right) at the point where x=3.

d

y = \ln \left(x^{4} + 4\right) at the point where x=2.

e

y = \ln \left( \sqrt x\right) at the point where x=e^2.

f

y = \ln \left(x - 3\right) at the point where x=4.

g

y = \ln \left( 3 x - 2\right) at the point where x=1.

h

y = x^{2} \ln x^{2} at the point where x = e.

12

At the point \left(a, b\right) on the curve y = \ln \left( - 2 x\right), the gradient of the tangent to the curve is - \dfrac{1}{3}. Determine the value of a.

13

For each of the following functions:

i

State the domain of y.

ii

Find y \rq.

iii

State the domain of y \rq.

a
y = \ln \left( - 3 x \right)
b
y = \ln \left(x - 7\right) + \ln x
14

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.

15

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).

16

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.

Further differentiation
17

Differentiate the following functions:

a

y = 5\ln \sqrt{x}

b

y = \ln \left(\sqrt{ 7 x}\right)

c

y = \ln \left(\sqrt{2 - 5 x}\right)

d

y = \dfrac{\ln x}{x}

e

y = \dfrac{x}{\ln x}

f

y = \ln \left(\dfrac{8}{x + 4}\right)

g

y = \ln \left(\dfrac{4}{3 x + 4}\right)

h

y = \dfrac{\ln 3 x}{5 x^{4} + 2}

i

f \left( x \right) = \ln \left(\left( 4 x^{3} + 8 x^{2} - 9\right)^{3}\right)

j

y = 6 x^{4} \ln x

k

y = x \ln x

l

y = \left(\ln x\right)^{6}

m

y = 3 \ln x + 5 \ln 2 x

n

y = x^{4} + 6 \ln 7 x

o

y = 4 \ln x - \dfrac{1}{x}

p

y = \ln \left(\dfrac{3}{x}\right)

q

y = \left(x + 6\right) \ln \left(x + 6\right)

r

y = x^{5} \ln \left(x - 5\right)

s

y = \left(x^{2} + 6\right) \ln 2 x

18

Differentiate the following functions:

a

y = \dfrac{\ln x}{\sin x}

b

y = \cos \left(\ln x\right)

c

y = \cos x \ln x

d

y = \ln e^{ 8 x}

e

y = e^{\ln x}

f

y = \log_{2} x

g

y = e^{x} \ln \left(x\right)

h

y = \dfrac{\ln x}{e^{x}}

i

y = \dfrac{\ln x}{e^{ 2 x}}

j

y = e^{ - 2 } \ln \left( - 6 + x^{ - 3 }\right)

k

y = \ln \left(\log_{e} 4 x\right)

l
y = e^{ 2 x} \ln \left( 4 x\right)
m

y = \ln \left(\ln x\right)

19

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}.

20

Consider the function y = x e^{x}.

a

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

b

Hence, find \dfrac{d y}{d x}, without using the product rule.

21

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.

22

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).

23

Determine \dfrac{d y}{d x}, given that y = u^{5} and u = \ln \left(x + 5\right).

24

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}.

25

Consider the function y = \ln \left(\ln x \right).

a

Find the derivative of y.

b

Find y when x = e.

c

Hence, find the equation of the tangent to the curve at x = e.

26

Consider the function y = 2 \ln \left(x^{2} + e \right).

a

Find the derivative of y.

b

Evaluate the derivative at x = 0.

c

Hence, find the equation of the tangent to the curve at x = 0.

27

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}.

28

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}.

29

Consider the function y = \ln \left(\dfrac{1}{\left(x - 4\right)^{3}}\right).

a

Simplify the function.

b

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

c

State the values of x for which the function is undefined.

d

Determine whether the function is strictly increasing, strictly decreasing, or neither. Explain your answer.

30

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

a

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

b

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

c

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

d

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

e

State whether the function is increasing or decreasing at x = 2.

f

Describe the concavity of the function at x = 2.

31

Consider the function f \left( x \right) = \dfrac{\ln x}{x} for x \gt 0.

a

Find the x-coordinate of the stationary point.

b

State whether the stationary point is a maximum or minimum.

32

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

a

State the domain of this function.

b

Find the exact coordinates of the turning point.

c

Determine the nature of the turning point.

33

Consider the function y = x \ln x.

a

State the domain of the function.

b

Find y \rq.

c

Determine the exact coordinates of the turning point.

d

Find y \rq \rq at the turning point.

e

State whether the turning point is a maximum or minimum value of the function.

f

State the range of the function in exact form.

g

Describe the behaviour of the function as x \to \infty.

h

Sketch the graph of the function.

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establish and use the formula d/dx(ln⁡x)=1/x

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