topic badge

4.01 Differentiation Review

Worksheet
Differentiation of individual terms
1

Differentiate the following:

a
y=6x^3
b
y = 8x^4+7
c
f(x)=10x^2-9x+3
d
f(x)=7x^4 - 9x^2
e
f(x)=5x^2-8x+10-\dfrac{3}{x}
f
y = 12x^{\frac{1}{2}}
g
y=\dfrac{x^2-8x+1}{x}
h
y=8x^2 - 10x^{-2}+x^{-5}
i
y = x^{\frac{1}{2}} + 8 x^{\frac{3}{4}}
j
y = \left( 5 x\right)^{3} + 6 \sqrt[3]{x}
k
f(x)=\sqrt{x}
l
y=\dfrac{11}{x^3} - \sqrt[5]{x^4}
2

Find the derivative of y = x^{6} + x^{\frac{1}{5}}. Write your answer with positive indices.

3

Differentiate y = 18 x \sqrt{x}. Leave your answer in surd form.

Chain rule
4

Let y = \left( 4 x + 3\right)^{ - 1 } be defined as a composition of the functions y = u^{ - 1 } and u = 4 x + 3. Find:

a

\dfrac{d y}{d u}

b

\dfrac{d u}{d x}

c

\dfrac{dy}{dx}

5

Consider the function y = \left( 2 x^{4} + 6\right)^{5} where u = 2 x^{4} + 6.

a

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

b

Express y as a function of u.

c

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

d

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

6

Consider the function f \left( x \right) = \left( 5 x^{3} - 4 x^{2} + 3 x - 5\right)^{7}.

a

Redefine the function as composite functions f \left( u \right) and u \left( x \right), where u \left( x \right) is a polynomial.

b

Differentiate f(x) using the chain rule.

7

Differentiate each of the following:

a
y = x^{\frac{5}{3}} + \left( 4 x - 7\right)^{6}
b
y = \left(x + 7\right)^{5} - \dfrac{3}{x^{4}}
c
y = x^{4} - 4 x + 6 \sqrt{x - 7}
d
y = x^{7} + 3 x^{4} + \sqrt[3]{x + 4}
e
y = \dfrac{3}{x + 7} + \dfrac{6 x + 7}{3}
f
y = \left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)^{8}
g
y = \dfrac{1}{\sqrt[3]{\left(9 - x\right)^{4}}}
8

Differentiate y = 4 x^{5} + \dfrac{1}{x - 4} + 6. Express the answer using positive indices.

Product rule
9

Consider the function y = x^{6} \left(x^{4} + 4\right). Let u = x^{6} and v = x^{4} + 4. Find:

a

u'

b

v'

c

\dfrac{dy}{dx}

10

Differentiate the following using the product rule:

a
y = x^{3} \left( 5 x + 3\right)^{7}
b
y = 6 x^{5} \left(x^{2} + 3\right)^{3}
c
y = 3 x \left(x^{2} + x + 1\right)^{9}
d
g \left( t \right) = \left( 3 t^{3} + 4\right) \left(1 - t\right)
e
y = 8 x \left(5 + 8 x\right)^{\frac{7}{4}} - 3
f
y = 8 x^{5} \sqrt{ 8 x + 3}
g
g \left( t \right) = \left( 2 t^{3} - 3\right) \left(3 - t\right)
h
y = \left( 8 x - 9\right)^{5} \left( 5 x + 7\right)^{7}
11

Differentiate y = \left( 3 x + 2\right) \sqrt{5 + 4 x}. Express your answer as a single fraction.

12

For each of the following equations:

i

Differentiate the function.

ii

State the values of x for which the derivative is zero.

a
y = \left(x + 2\right) \left(x + 5\right)^{6}
b
y = x \left(x - 8\right)^{4}
Quotient rule
13

Consider the function y = \dfrac{2 x - 5}{5 x - 2}.

a

Using the substitution u = 2 x - 5, find u'.

b

Using the substitution v = 5 x - 2, find v'.

c

Hence find y'.

d

Is it possible for the derivative of this function to be zero?

14

Differentiate the following functions using the quotient rule:

a
f(x)=\dfrac{7x}{8x-1}
b
y=\dfrac{4x^2}{3x-7}
c
y = \dfrac{3 x}{5 x - 4}
d
y = \dfrac{3 x^{2} + 2}{5 x^{2} + 4}
e
f(x)=\dfrac{2x+1}{x^2 -6x}
f
y=\dfrac{(x-5)^3}{8x}
g
f(x)=\dfrac{\sqrt{x-10}}{4x^2-6}
h
y=\dfrac{(2x-3)^4}{\sqrt{2x+5}}
i
y = \sqrt{\dfrac{2 + 7 x}{2 - 7 x}}
j
f \left( t \right) = \dfrac{\left( 4 t^{2} + 3\right)^{3}}{\left(5 + 2 t\right)^{5}}
15

Differentiate y = \dfrac{x^{2}}{x + 3} and find the value of a if y' = 0 at x = a.

16

Find the value of f' \left( 0 \right) if f \left( x \right) = \dfrac{x}{\sqrt{16 - x^{2}}}.

Applications
17

Find the gradient of f \left( x \right) = \left(x - 8\right)^{3} \left(x - 4\right)^{4} at x = 6.

18

Find the gradient of f \left( x \right) = \dfrac{\left(x + 7\right)^{9}}{\left(x + 9\right)^{4}} at x = - 5.

19

Find the equation of the normal to the curve y = \left(x^{2} + 1\right)^{4} at the point where x = 1.

20

Find the values of x such that the gradient of the tangent to the curve y = \dfrac{6 x - 1}{3 x - 1} is - 3.

21

Consider the function f \left( x \right) = \dfrac{4 x^{9}}{\left(x + 2\right)^{4}}.

a

Find f' \left( 2 \right).

b

Is the function increasing or decreasing at x = 2?

22

The curve y = \sqrt{x - 3} has a tangent with a gradient of \dfrac{1}{2} at the point P. Find the coordinates of P.

23

f \left( x \right) and g \left( x \right) are differentiable functions such that f \left( x \right) \neq g \left( x \right).

Suppose f \left( 2 \right) = 3, g \left( 2 \right) = 2, f' \left( 2 \right) = 4 and g' \left( 2 \right) = 5.

a

Find the value of \dfrac{d}{dx} \left( f \left( x \right) g \left( x \right)^{4} \right) at x = 2.

b

Find the value of \dfrac{d}{dx} \left(\dfrac{g \left( x \right)}{f \left( x \right) + g \left( x \right)}\right) at x = 2.

Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

MA12-3

applies calculus techniques to model and solve problems

MA12-6

applies appropriate differentiation methods to solve problems

What is Mathspace

About Mathspace