A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 9: Differentiation · Lesson 1

Differentiating sin x, cos x, eˣ and ln x

116 marks

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Date

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Quick questions to start.

A1
Needed today
Differentiate $y = 4 x^{3} - \frac{2}{x} + 6 x$ .
[3 marks]
Answer
A2
Needed today
Expand $sin (x + h)$ and $cos (x + h)$ .
[2 marks]
Answer
A3
Common slip
Write $4 5^{\circ}$ and $15 0^{\circ}$ in radians. Say why the rule $\frac{d}{d x} (sin x) = cos x$ needs $x$ in radians.
[2 marks]
Answer
A4
Needed today
Write $ln (5 x)$ as a sum of two logarithms, and write $e^{2 x + 3}$ as a constant times $e^{2 x}$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Prove, from first principles, that the derivative of $cos x$ is $- sin x$ . You may use that, as $h \to 0$ , $\frac{sin h}{h} \to 1$ and $\frac{cos h - 1}{h} \to 0$ .

Hint Write the gradient of the chord from x to x + h, expand cos(x + h), and group the terms with cos x and with sin x.

B1
Prove, from first principles, that the derivative of $sin 3 x$ is $3 cos 3 x$ . You may use that, as $h \to 0$ , $\frac{sin h}{h} \to 1$ and $\frac{cos h - 1}{h} \to 0$ .
[5 marks]

Example 2

The curve $C$ has equation $y = 2 sin 3 x + 4 cos \frac{x}{2}$ . Find the exact gradient of $C$ at the point where $x = \frac{π}{3}$ .

Hint sin kx gives k cos kx and cos kx gives −k sin kx. Then put in x = π/3, in radians.

B2
The curve $C$ has equation $y = 5 cos 2 x - 3 sin \frac{x}{3}$ . Find an equation of the tangent to $C$ at the point where $x = \frac{π}{2}$ , giving exact values.
[6 marks]
Answer

Example 3

Given $y = 4 e^{\frac{x}{2}} - 6 ln (3 x)$ , find the exact value of $\frac{d y}{d x}$ when $x = 2 ln 2$ .

Hint e to the kx gives k e to the kx. ln(3x) = ln 3 + ln x, so its derivative is 1/x.

B3
Given $y = 2 e^{3 x} + 5 ln (2 x)$ , find the exact value of $\frac{d y}{d x}$ when $x = \frac{1}{3} ln 2$ .
[4 marks]
Answer

Example 4

By writing $2^{3 x}$ as a power of $e$ , show that $\frac{d}{d x} (2^{3 x}) = 3 ln 2 \times 2^{3 x}$ . Find the gradient of $y = 2^{3 x}$ at $x = 1$ .

Hint Write

2 = e^{l n 2}

, so

2^{3 x} = e^{(3 l n 2) x}

. Then use the rule for

e^{k x}

with

k = 3 ln 2

B4
The curve $y = 5^{2 x}$ passes through the point $P (\frac{1}{2}, 5)$ . Find an equation of the tangent to the curve at $P$ , giving exact values.
[5 marks]
Answer

Example 5

Find the exact coordinates of the stationary point on the curve $y = e^{2 x} - 6 x$ , and decide whether it is a maximum or a minimum.

Hint Set the derivative equal to

0

and solve for

e^{2 x}

first. Then take logs.

B5
Find the exact coordinates of the stationary point on the curve $y = 4 x - e^{\frac{x}{2}}$ , and decide whether it is a maximum or a minimum.
[6 marks]
Answer

Practice

A-Level

C1
Mia differentiates $y = 3 cos 4 x + ln (7 x)$ and writes $\frac{d y}{d x} = 3 sin 4 x + \frac{7}{x}$ . Find her two mistakes and give the correct derivative.
[4 marks]
Answer

C2
Liam is asked for the gradient of $y = sin x^{\circ}$ at $x = 60$ , where $x$ is in degrees. He writes " $cos 6 0^{\circ} = \frac{1}{2}$ ". Explain why Liam is wrong, and find the correct gradient.
[4 marks]
Answer
C3
Ben says the derivative of $3^{x}$ is $x \times 3^{x - 1}$ , "the same as for $x^{n}$ ". Explain why Ben is wrong. Find the exact gradient of $y = 3^{x}$ at $x = 2$ and compare it with Ben's value.
[4 marks]
Answer
C4
Find the exact coordinates of the points on the curve $y = sin x + 3 cos x$ , $0 \leq x < 2 π$ , where the gradient is zero.
[5 marks]
Answer
C5
The tangent to the curve $y = ln x$ at the point where $x = a$ passes through the origin. Find $a$ and the equation of the tangent.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The curve $C$ has equation $y = e^{2 x} - 5 e^{x} + 3 x$ .
(a)
Find $\frac{d y}{d x}$ .
[2 marks]
Answer
(b)
Hence find the exact coordinates of the two stationary points of $C$ .
[5 marks]
Answer
(c)
Determine the nature of each stationary point.
[4 marks]
Answer
Total for D1: 11 marks

D2
Priya is finding the normal to $y = 4 cos 2 x + 1$ at the point where $x = \frac{π}{6}$ . She writes: " $\frac{d y}{d x} = 8 sin 2 x$ , so the gradient at $x = \frac{π}{6}$ is $43$ . The point is $(\frac{π}{6}, 3)$ , so the normal is $y - 3 = 43 (x - \frac{π}{6})$ ."
Priya is wrong.
(a)
Identify her two mistakes.
[3 marks]
(b)
Find the correct equation of the normal in the form $y = m x + c$ , giving $m$ and $c$ exactly.
[4 marks]
Answer
(c)
Find the exact $x$ -coordinate of the point where the normal meets the $x$ -axis.
[3 marks]
Answer
Total for D2: 10 marks
D3
The tangent to the curve $y = e^{3 x} - 9 x$ at the point $P$ is parallel to the line $y = 3 x + 1$ . Find the exact coordinates of $P$ and the equation of this tangent, and show that the tangent meets the $y$ -axis below the origin.
[8 marks]
Answer
D4
The temperature, $T^{\circ} C$ , of a cup of tea $t$ minutes after it is made is modelled by $T = 20 + 60 e^{- 0.05 t}$ , $t \geq 0$ .
(a)
State the temperature of the tea when it is made.
[2 marks]
Answer
(b)
Find the rate of change of the temperature when $t = 10$ . Give your answer to $3$ significant figures, with units.
[3 marks]
Answer
(c)
Find the exact time at which the tea is cooling at $1.5^{\circ} C$ per minute, and give it to the nearest tenth of a minute.
[4 marks]
Answer
(d)
Give one feature of the model that suggests the room is at a temperature of 20 degrees, and one reason the model may not fit for large values of t.
[2 marks]
Total for D4: 11 marks
D5
The function f is defined by $f (x) = 2 x - ln (4 x)$ , $x > 0$ .
(a)
Find $f^{'} (x)$ .
[2 marks]
Answer
(b)
Find the exact coordinates of the stationary point of $y = f (x)$ .
[3 marks]
Answer
(c)
Show that this stationary point is a minimum.
[2 marks]
(d)
Hence show that $ln (4 x) \leq 2 x - 1 + ln 2$ for all $x > 0$ .
[3 marks]
Total for D5: 10 marks

Extension

Stretch

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

E1
The line $y = m x$ is a tangent to the curve $y = ln (2 x)$ . Find the exact value of $m$ . Hence find the set of values of $k$ for which the equation $ln (2 x) = k x$ has exactly two real solutions.
[9 marks]
Hint 1 · What to try
Let the tangent touch the curve at $x = a$ . Write its equation using the gradient $\frac{1}{a}$ .
Hint 2 · The first line
The tangent passes through $(0, 0)$ . Put that in to find $a$ , then $m$ .
Hint 3 · The full method
Sketch $y = ln (2 x)$ and the lines $y = k x$ through the origin. How many crossings for $k$ below $m$ , equal to $m$ , above $m$ , and for $k \leq 0$ ?
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can prove the derivative of sin kx or cos kx from first principles.
I can differentiate sin kx, cos kx, e to the kx, ln kx and a to the x, with x in radians.
I can find gradients, tangents and normals to curves with these functions, giving exact answers.
I can find and classify stationary points of curves with exponentials and logarithms.

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