A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 9: Differentiation · Lesson 4

Differentiating trigonometric functions

117 marks

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Quick questions to start.

A1
Last lesson
Differentiate $y = x e^{3 x}$ .
[2 marks]
Answer
A2
Needed today
Given that $tan x = \frac{3}{4}$ and $x$ is acute, find the exact value of $sec x$ .
[2 marks]
Answer
A3
Needed today
Write $y = \frac{1}{sin x}$ as $(sin x)^{- 1}$ and use the chain rule to find $\frac{d y}{d x}$ .
[3 marks]
Answer
A4
Common slip
Sam writes that the derivative of $cos 3 x$ is $3 sin 3 x$ . Explain his mistake and give the correct derivative.
[2 marks]

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Prove that $\frac{d}{d x} (tan x) = sec^{2} x$ .

Hint Write tan x as sin x over cos x and use the quotient rule. Then use sin squared plus cos squared equals 1.

B1
Prove that $\frac{d}{d x} (cot x) = - cosec^{2} x$ .
[4 marks]

Example 2

Show that the derivative of $sec x$ is $sec x tan x$ .

Hint Write sec x as cos x to the power minus 1 and use the chain rule.

B2
Given $y = cosec 3 x$ , find $\frac{d y}{d x}$ and the exact gradient of the curve at the point where $x = \frac{π}{12}$ .
[4 marks]
Answer

Example 3

Given $y = x tan 2 x$ , find $\frac{d y}{d x}$ .

Hint Product rule, with the chain rule giving 2 sec squared 2x for the tan.

B3
The curve $C$ has equation $y = e^{2 x} sec x$ , $- \frac{π}{2} < x < \frac{π}{2}$ . Find an equation of the tangent to $C$ at the point where $x = 0$ .
[5 marks]
Answer

Example 4

Given that $x = sin y$ , $- \frac{π}{2} < y < \frac{π}{2}$ , find $\frac{d y}{d x}$ in terms of $x$ .

Hint Find dx/dy, turn it upside down, then write cos y in terms of x. In this range cos y is positive.

B4
Given that $x = tan y$ , $- \frac{π}{2} < y < \frac{π}{2}$ , show that $\frac{d y}{d x} = \frac{1}{1 + x ^{2}}$ .
[4 marks]

Example 5

Find the exact $x$ -coordinates of the stationary points of $y = 4 x - tan x$ , $- \frac{π}{2} < x < \frac{π}{2}$ .

Hint Set 4 minus sec squared x equal to 0, so cos squared x is one quarter. Take both signs of the square root.

B5
The curve $C$ has equation $y = cot x + 2 x$ , $0 < x < π$ . Find the exact $x$ -coordinates of the stationary points of $C$ and decide the nature of each.
[5 marks]
Answer

Practice

A-Level

C1
Tom differentiates $y = tan^{3} 2 x$ and writes $\frac{d y}{d x} = 3 tan^{2} 2 x sec^{2} 2 x$ . Explain his mistake, then find the gradient of the curve at the point where $x = \frac{π}{8}$ .
[4 marks]
Answer

C2
The curve $C$ has equation $y = ln (sec x)$ , $0 < x < \frac{π}{2}$ . Show that $\frac{d y}{d x} = tan x$ , and find an equation of the tangent to $C$ at the point where $x = \frac{π}{4}$ , in exact form.
[5 marks]
Answer
C3
Given that $x = cot y$ , $0 < y < π$ , show that $\frac{d y}{d x} = - \frac{1}{1 + x ^{2}}$ .
[4 marks]
C4
Given $y = \frac{sin x}{1 + cos x}$ , find $\frac{d y}{d x}$ in its simplest form.
[5 marks]
Answer
C5
Find the exact coordinates of the stationary point on the curve $y = 2 sec x - tan x$ , $- \frac{π}{2} < x < \frac{π}{2}$ .
[6 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Kai says: "The derivative of $sec^{2} x$ is $2 sec x tan x$ , because the derivative of $sec x$ is $sec x tan x$ ."
Kai is wrong.
(a)
Explain his mistake and find the correct derivative of $sec^{2} x$ .
[3 marks]
Answer
(b)
Use the identity $tan^{2} x = sec^{2} x - 1$ to check your answer, by differentiating $tan^{2} x$ .
[3 marks]
(c)
Find the exact gradient of the curve $y = sec^{2} x$ at the point where $x = \frac{π}{3}$ .
[3 marks]
Answer
Total for D1: 9 marks

D2
The curve $C$ has equation $y = e^{- x} tan x$ , $- \frac{π}{2} < x < \frac{π}{2}$ .
(a)
Show that $\frac{d y}{d x} = e^{- x} (tan^{2} x - tan x + 1)$ .
[3 marks]
(b)
Hence show that $C$ has no stationary points.
[4 marks]
(c)
Find an equation of the tangent to $C$ at the origin.
[2 marks]
Answer
(d)
Find the exact gradient of $C$ at the point where $x = \frac{π}{4}$ .
[2 marks]
Answer
Total for D2: 11 marks
D3
Find an equation of the normal to the curve $y = cosec x$ at the point where $x = \frac{π}{6}$ , in the form $y = m x + c$ , where $m$ and $c$ are exact.
[6 marks]
Answer
D4
The curve $C$ has equation $x = sec y$ , $0 < y < \frac{π}{2}$ .
(a)
Show that $\frac{d y}{d x} = \frac{1}{x x ^{2} - 1}$ .
[4 marks]
(b)
Find an equation of the tangent to $C$ at the point where $y = \frac{π}{3}$ .
[3 marks]
Answer
(c)
Explain why the gradient of $C$ is positive at every point.
[2 marks]
Total for D4: 9 marks
D5
A straight rod touches the inside corner where a corridor $3$ m wide meets a corridor $4$ m wide at right angles, and its ends touch the two outer walls. When the rod makes an angle $θ$ with the outer wall of the wider corridor, its length is $L = 3 sec θ + 4 cosec θ$ , $0 < θ < \frac{π}{2}$ . The shortest such rod gives the longest rod that can be carried round the corner flat.
(a)
Find $\frac{d L}{d θ}$ .
[2 marks]
Answer
(b)
Show that $L$ is stationary when $tan^{3} θ = \frac{4}{3}$ .
[3 marks]
(c)
Find this value of $θ$ , in radians to 3 significant figures.
[2 marks]
Answer
(d)
Find the length of the longest rod that can be carried round the corner flat, to 3 significant figures.
[2 marks]
Answer
(e)
Show that this value of $θ$ gives the least value of $L$ .
[2 marks]
Total for D5: 11 marks
D6
The curve $C$ has equation $y = ln (tan x)$ , $0 < x < \frac{π}{2}$ .
(a)
Show that $\frac{d y}{d x} = 2 cosec 2 x$ .
[4 marks]
(b)
Find the least value of the gradient of C, and the exact value of x at which it occurs.
[3 marks]
Answer
Total for D6: 7 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The curve $C$ has equation $y = sec x + k tan x$ , $- \frac{π}{2} < x < \frac{π}{2}$ , where $k$ is a constant. Show that $C$ has a stationary point exactly when $- 1 < k < 1$ , that this stationary point is always a minimum, and that the least value of $y$ is $1 - k^{2}$ .
[9 marks]
Hint 1 · What to try
Differentiate, then write the derivative over the single denominator $cos^{2} x$ . When is the top zero?
Hint 2 · The first line
For which values of k does sin x equal minus k for an x strictly between minus a right angle and a right angle?
Hint 3 · The full method
At the stationary point $sin x = - k$ . Find $cos x$ in terms of $k$ , then write $y$ over $cos x$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can differentiate tan kx, sec kx, cosec kx and cot kx.
I can prove these results with the quotient rule or the chain rule.
I can combine them with the chain, product and quotient rules.
I can find dy/dx in terms of x when x is a trigonometric function of y.

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