Print the chapter

Every lesson of the chapter, ready for paper. Exercise sets are lettered A, B, C… and answered at the end.

Include answers

To save a PDF, choose “Save as PDF” in the print dialog.

1. The power, constant multiple and sum rules

Warm-up

Three quick questions from earlier lessons.

Discover

From the limit definition: $(x^2)' = 2x$ and $(x^3)' = 3x^2$ . Guess the pattern: what is $(x^4)'$ ?

$4x^3$
$x^3$

Key idea

Three rules replace the limit for every polynomial and power.

Power rule:

(x^n)' = n\,x^{n-1}

for every real

n

. Constant multiple:

(k\,f)' = k\,f'

. Sum:

(f + g)' = f' + g'

. Constant:

(c)' = 0

Worked example

$f(x) = 5x^3 - \dfrac{2}{x} + 7$ . Find $f'(x)$ .

f(x) = 5x^3 - 2x^{-1} + 7

Write the fraction as a power.

f'(x) = 15x^2 + 2x^{-2}

Power rule term by term; the constant 7 gives 0.

f'(x) = 15x^2 + \dfrac{2}{x^2}

Back to a fraction if you like.

Your turn, guided

Your turn: $f(x) = 4x^5 + 3x - 9$ .

f'(x) = 20x^4 + 3

20x⁴ from 4x⁵, 3 from 3x, 0 from −9.

Practice ★

$f(x) = 8x^{ 5 } + 3x$ . Find $f'(x)$ .
$f(x) = 4x^{ 3 } + 6x$ . Find $f'(x)$ .
$f(x) = 6x^{ 5 } + 5x$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(x) = \dfrac{ 6 }{x^{ 2 } }$ . Find $f'(x)$ .
$f(x) = \dfrac{ 9 }{x^{ 3 } }$ . Find $f'(x)$ .

$f(x) = 3\sqrt{x} + 8$ . Find $f'(x)$ .
$f(x) = 5\sqrt{x} + 4$ . Find $f'(x)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 8x^{ 5 } + 3x$ . Find $f'(x)$ .
$f(x) = 4x^{ 3 } + 6x$ . Find $f'(x)$ .
$f(x) = 6x^{ 5 } + 5x$ . Find $f'(x)$ .

Summary card

Basic rules

$(x^n)' = n x^{n-1}$ ; rewrite roots and fractions as powers first.
Constants factor out; sums split; a constant alone gives 0.

Watch out: with negative exponents, $-2 - 1 = -3$ .

2. Derivatives of exponential functions

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curve is $y = e^x$ , the blue line its tangent at $x = a$ . Move $a$ and compare the slope of the line with the height of the point.

The curve y = eˣ and its tangent at x = a. The tangent's slope equals the height eᵃ of the point of contact.

At $a = 0$ the point is $(0, 1)$ . What is the slope of the tangent there?

Key idea

The exponent varies, so the power rule does not apply.

(e^x)' = e^x

. For any base

b > 0

(b^x)' = b^x \ln b

Worked example

$f(x) = 3e^x - 2^x$ . Find $f'(x)$ .

(3e^x)' = 3e^x

eˣ is its own derivative.

(2^x)' = 2^x \ln 2

Base 2: multiply by ln 2.

f'(x) = 3e^x - 2^x \ln 2

Combine.

Your turn, guided

Your turn: $f(x) = 5e^x + x^3$ .

f'(x) = 5e^x + 3x^2

5eˣ stays; x³ gives 3x².

Practice ★

$f(x) = 8e^x + 5x^2$ . Find $f'(x)$ .
$f(x) = 4e^x + 6x^2$ . Find $f'(x)$ .
$f(x) = 7e^x + 4x^2$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(x) = 3^x$ . Find $f'(x)$ .
$f(x) = 4^x$ . Find $f'(x)$ .

Find the slope of the tangent line to $y = 8e^x + 1$ at $x = 0$ .
Find the slope of the tangent line to $y = 2e^x + 4$ at $x = 0$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 3^x$ . Find $f'(x)$ .
$f(x) = 4^x$ . Find $f'(x)$ .
$f(x) = 2^x$ . Find $f'(x)$ .

Summary card

Exponentials

$(e^x)' = e^x$ .
$(b^x)' = b^x \ln b$ .

Watch out: $x^n$ (variable base) and $b^x$ (variable exponent) have different rules.

3. Derivatives of sine and cosine

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curves are $y = \sin x$ and $y = \cos x$ ; the blue line is the tangent to $\sin x$ at $x = a$ . Move $a$ and compare the slope of the line with the height of the cosine curve at $x = a$ .

The sine curve, the cosine curve and the tangent to the sine curve at x = a. The tangent's slope is always the height of the cosine curve at a.

At $a = \pi/2 \approx 1.57$ , sine is at its top. What is the slope there, and what is $\cos(\pi/2)$ ?

Both are 0
Both are 1

Key idea

With $x$ in radians:

(\sin x)' = \cos x

and

(\cos x)' = -\sin x

Worked example

$f(x) = 3\sin x - 4\cos x$ . Find $f'(x)$ .

(3\sin x)' = 3\cos x

(sin x)' = cos x.

(-4\cos x)' = 4\sin x

(cos x)' = −sin x, and − · − = +.

f'(x) = 3\cos x + 4\sin x

Combine.

Your turn, guided

Your turn: $f(x) = 2\cos x + x^2$ .

f'(x) = -2\sin x + 2x

−2 sin x and 2x.

Practice ★

$f(x) = 3\sin x + 5\cos x$ . Find $f'(x)$ .
$f(x) = 5\sin x + 7\cos x$ . Find $f'(x)$ .
$f(x) = 5\sin x + 9\cos x$ . Find $f'(x)$ .

Watch out!

Practice ★★

Find the slope of the tangent line to $y = \sin x$ at $x = \dfrac{\pi}{ 2 }$ . Give an exact value.
Find the slope of the tangent line to $y = \sin x$ at $x = \dfrac{\pi}{ 3 }$ . Give an exact value.

$f(x) = 4x - 4\cos x$ . Find $f'(x)$ .
$f(x) = 4x - 8\cos x$ . Find $f'(x)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 3\sin x + 5\cos x$ . Find $f'(x)$ .
$f(x) = 5\sin x + 7\cos x$ . Find $f'(x)$ .
$f(x) = 5\sin x + 9\cos x$ . Find $f'(x)$ .

Summary card

Sine and cosine

$(\sin x)' = \cos x$ .
$(\cos x)' = -\sin x$ .

Watch out: the minus sign goes with the derivative of cosine; radians only.

4. The product rule

Warm-up

Three quick questions from earlier lessons.

Discover

$f(x) = x$ and $g(x) = x$ , so $f(x)g(x) = x^2$ . Is $(x^2)'$ equal to $f'(x)\,g'(x) = 1 \cdot 1$ ?

Key idea

Each factor takes its turn to be differentiated.

Product rule:

(f\,g)' = f'\,g + f\,g'

Worked example

$h(x) = x^3 e^x$ . Find $h'(x)$ .

f' = 3x^2

g' = e^x

f = x³, g = eˣ.

h'(x) = 3x^2 e^x + x^3 e^x

f'g + fg'.

h'(x) = x^2 e^x (3 + x)

Factor if you like.

Your turn, guided

Your turn: $h(x) = x\cos x$ .

f' = 1

g' = -\sin x

f = x, g = cos x.

h'(x) = \cos x - x\sin x

f'g + fg'.

Practice ★

$f(x) = x^{ 2 }\sin x$ . Find $f'(x)$ .
$f(x) = x^{ 3 }\sin x$ . Find $f'(x)$ .

$f(x) = (x + 3)e^x$ . Find $f'(x)$ .
$f(x) = (x + 8)e^x$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(2) = 4$ , $f'(2) = 4$ , $g(2) = -4$ and $g'(2) = -1$ . Find $(fg)'(2)$ .
$f(2) = -4$ , $f'(2) = -2$ , $g(2) = 2$ and $g'(2) = 3$ . Find $(fg)'(2)$ .
$f(2) = -1$ , $f'(2) = 1$ , $g(2) = 5$ and $g'(2) = -1$ . Find $(fg)'(2)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = x^{ 2 }\sin x$ . Find $f'(x)$ .
$f(x) = x^{ 3 }\sin x$ . Find $f'(x)$ .
$f(x) = x^{ 5 }\sin x$ . Find $f'(x)$ .

Summary card

Product rule

$(fg)' = f'g + fg'$ : differentiate one factor at a time, keep the other.

Watch out: $(fg)' \ne f'g'$ .

5. The quotient rule

Warm-up

Three quick questions from earlier lessons.

Discover

$\dfrac{x^3}{x} = x^2$ , whose derivative is $2x$ . The quotient of the derivatives would be $\dfrac{3x^2}{1}$ . Do they agree?

Key idea

Top's derivative first, then subtract.

Quotient rule:

\left(\dfrac{f}{g}\right)' = \dfrac{f'\,g - f\,g'}{g^2}

, where

g \ne 0

Worked example

$h(x) = \dfrac{x^2}{x + 1}$ . Find $h'(x)$ .

f' = 2x

g' = 1

f = x², g = x + 1.

h'(x) = \dfrac{2x(x + 1) - x^2}{(x + 1)^2}

f'g − fg' over g².

h'(x) = \dfrac{x^2 + 2x}{(x + 1)^2}

Simplify the top.

Your turn, guided

Your turn: $h(x) = \dfrac{e^x}{x}$ .

h'(x) = \dfrac{xe^x - e^x}{x^2}

f = eˣ, g = x: f'g − fg' = xeˣ − eˣ.

Practice ★

$f(x) = \dfrac{x}{x + 6}$ . Find $f'(x)$ .
$f(x) = \dfrac{x}{x + 3}$ . Find $f'(x)$ .
$f(x) = \dfrac{x}{x + 2}$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(x) = \dfrac{ 4\sin x}{x}$ . Find $f'(x)$ .
$f(x) = \dfrac{ 1\sin x}{x}$ . Find $f'(x)$ .

$f(1) = 1$ , $f'(1) = -2$ , $g(1) = 4$ and $g'(1) = 2$ . Find $\left(\dfrac{f}{g}\right)'(1)$ .
$f(1) = 1$ , $f'(1) = 1$ , $g(1) = 1$ and $g'(1) = 4$ . Find $\left(\dfrac{f}{g}\right)'(1)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = \dfrac{x}{x + 6}$ . Find $f'(x)$ .
$f(x) = \dfrac{x}{x + 3}$ . Find $f'(x)$ .
$f(x) = \dfrac{x}{x + 2}$ . Find $f'(x)$ .

Summary card

Quotient rule

$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$ : "low d-high minus high d-low, over low squared".

Watch out: the order in the numerator matters.

6. Derivatives of tangent, secant, cotangent and cosecant

Warm-up

Three quick questions from earlier lessons.

Discover

$\tan x = \dfrac{\sin x}{\cos x}$ . With the quotient rule, the top of $(\tan x)'$ is $\cos x \cdot \cos x - \sin x \cdot (-\sin x)$ . What does it simplify to?

1
$\cos^2 x - \sin^2 x$

Key idea

The quotient rule gives the other four.

(\tan x)' = \sec^2 x

(\sec x)' = \sec x \tan x

(\cot x)' = -\csc^2 x

(\csc x)' = -\csc x \cot x

. The "co" functions get a minus sign.

Worked example

$f(x) = 2\tan x - \csc x$ . Find $f'(x)$ .

(2\tan x)' = 2\sec^2 x

(tan x)' = sec² x.

(-\csc x)' = \csc x \cot x

(csc x)' = −csc x cot x, and − · − = +.

f'(x) = 2\sec^2 x + \csc x\cot x

Combine.

Your turn, guided

Your turn: $f(x) = \sec x + x$ .

f'(x) = \sec x\tan x + 1

sec x tan x, plus 1.

Practice ★

$f(x) = 7\tan x$ . Find $f'(x)$ .
$f(x) = 3\tan x$ . Find $f'(x)$ .
$f(x) = 5\tan x$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(x) = \sec x + 2\cot x$ . Find $f'(x)$ .
$f(x) = \sec x + 8\cot x$ . Find $f'(x)$ .

Find the slope of the tangent line to $y = \tan x$ at $x = \dfrac{\pi}{ 6 }$ . Give an exact value.
Find the slope of the tangent line to $y = \tan x$ at $x = \dfrac{\pi}{ 4 }$ . Give an exact value.

Exit ticket

Three questions to check you have got it.

$f(x) = 7\tan x$ . Find $f'(x)$ .
$f(x) = 3\tan x$ . Find $f'(x)$ .
$f(x) = 5\tan x$ . Find $f'(x)$ .

Summary card

tan, sec, cot, csc

$(\tan x)' = \sec^2 x$ , $(\sec x)' = \sec x\tan x$ .
$(\cot x)' = -\csc^2 x$ , $(\csc x)' = -\csc x\cot x$ .

Watch out: co-functions take a minus sign. You may also write $\sec^2 x$ as $\dfrac{1}{\cos^2 x}$ .

7. The chain rule

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curve is $y = \sin x$ ; the blue one is $y = \sin(kx)$ . Increase $k$ .

The curves y = sin x and y = sin(kx). With k = 2 the blue curve oscillates twice as fast, and its steepest slope is 2 instead of 1.

At $k = 2$ the blue curve goes twice as fast. At $x = 0$ , what is the slope of $y = \sin(2x)$ ?

Key idea

For a function inside a function: differentiate the outside, keep the inside, multiply by the derivative of the inside.

Chain rule:

\big(f(g(x))\big)' = f'(g(x)) \cdot g'(x)

Worked example

$h(x) = (x^2 + 1)^5$ . Find $h'(x)$ .

f(u) = u^5

g(x) = x^2 + 1

Outside u⁵, inside u = x² + 1.

h'(x) = 5(x^2 + 1)^4 \cdot 2x

f'(u) = 5u⁴ with the inside kept; g'(x) = 2x.

h'(x) = 10x(x^2 + 1)^4

Tidy.

Your turn, guided

Your turn: $h(x) = e^{3x}$ .

h'(x) = 3e^{3x}

Outside eᵘ, inside 3x: eᵘ times 3.

Practice ★

$f(x) = (6x + 7)^{ 4 }$ . Find $f'(x)$ .
$f(x) = (7x + 6)^{ 4 }$ . Find $f'(x)$ .
$f(x) = (3x + 5)^{ 4 }$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(x) = \sin(4x^2)$ . Find $f'(x)$ .
$f(x) = \sin(6x^2)$ . Find $f'(x)$ .

$f(x) = e^{ 5x^2}$ . Find $f'(x)$ .
$f(x) = e^{ 4x^2}$ . Find $f'(x)$ .

Boss problem. $h(x) = x^{ 3 }\sin(4x)$ . Find the slope of its graph at $x = \dfrac{\pi}{ 4 }$ .

Exit ticket

Three questions to check you have got it.

$f(x) = (6x + 7)^{ 4 }$ . Find $f'(x)$ .
$f(x) = (7x + 6)^{ 4 }$ . Find $f'(x)$ .
$f(x) = (3x + 5)^{ 4 }$ . Find $f'(x)$ .

Summary card

Chain rule

$\big(f(g(x))\big)' = f'(g(x))\,g'(x)$ : outside's derivative (inside kept) times inside's derivative.

Watch out: do not forget the factor $g'(x)$ .

8. Derivatives of inverse functions: ln x, arcsin x, arctan x

Warm-up

Three quick questions from earlier lessons.

Discover

$y = \ln x$ means $e^y = x$ . The slope of $e^y$ is $e^y = x$ . The graphs of $\ln$ and $\exp$ are reflections, so their slopes are reciprocal. What is $(\ln x)'$ ?

$\dfrac{1}{x}$
$\ln x$

Key idea

The derivatives of the inverse functions come from the chain rule.

(\ln x)' = \dfrac{1}{x}

(

x > 0

);

(\arcsin x)' = \dfrac{1}{\sqrt{1 - x^2}}

;

(\arctan x)' = \dfrac{1}{1 + x^2}

. With an inside function:

(\ln u)' = \dfrac{u'}{u}

Worked example

$h(x) = \ln(x^3 + 2)$ . Find $h'(x)$ .

(\ln u)' = \dfrac{u'}{u}

Inside u = x³ + 2, u' = 3x².

h'(x) = \dfrac{3x^2}{x^3 + 2}

Substitute.

Your turn, guided

Your turn: $h(x) = \arctan(x^2)$ .

h'(x) = \dfrac{2x}{1 + x^4}

u = x², u' = 2x; divide by 1 + u² = 1 + x⁴.

Practice ★

$f(x) = \ln(9x)$ for $x > 0$ . Find $f'(x)$ .
$f(x) = \ln(2x)$ for $x > 0$ . Find $f'(x)$ .

$f(x) = \ln(x^2 + 2)$ . Find $f'(x)$ .
$f(x) = \ln(x^2 + 8)$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(x) = \arctan(7x)$ . Find $f'(x)$ .
$f(x) = \arctan(2x)$ . Find $f'(x)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = \ln(x^2 + 2)$ . Find $f'(x)$ .
$f(x) = \ln(x^2 + 8)$ . Find $f'(x)$ .
$f(x) = \ln(x^2 + 1)$ . Find $f'(x)$ .

Summary card

Inverse functions

$(\ln x)' = \dfrac{1}{x}$ , $(\ln u)' = \dfrac{u'}{u}$ .
$(\arcsin x)' = \dfrac{1}{\sqrt{1 - x^2}}$ , $(\arctan x)' = \dfrac{1}{1 + x^2}$ .

Watch out: the chain rule still applies to the inside.

Answers: Differentiation rules

1. The power, constant multiple and sum rules

$40 x^{4} + 3$
$12 x^{2} + 6$
$30 x^{4} + 5$

$- \frac{12}{x^{3}}$
$- \frac{27}{x^{4}}$

$\frac{3}{2 \sqrt{x}}$
$\frac{5}{2 \sqrt{x}}$

$40 x^{4} + 3$
$12 x^{2} + 6$
$30 x^{4} + 5$

2. Derivatives of exponential functions

$10 x + 8 e^{x}$
$12 x + 4 e^{x}$
$8 x + 7 e^{x}$

$3^{x} \ln{\left(3 \right)}$
$4^{x} \ln{\left(4 \right)}$

$3^{x} \ln{\left(3 \right)}$
$4^{x} \ln{\left(4 \right)}$
$2^{x} \ln{\left(2 \right)}$

3. Derivatives of sine and cosine

$- 5 \sin{\left(x \right)} + 3 \cos{\left(x \right)}$
$- 7 \sin{\left(x \right)} + 5 \cos{\left(x \right)}$
$- 9 \sin{\left(x \right)} + 5 \cos{\left(x \right)}$

$0$
$\frac{1}{2}$

$4 \sin{\left(x \right)} + 4$
$8 \sin{\left(x \right)} + 4$

$- 5 \sin{\left(x \right)} + 3 \cos{\left(x \right)}$
$- 7 \sin{\left(x \right)} + 5 \cos{\left(x \right)}$
$- 9 \sin{\left(x \right)} + 5 \cos{\left(x \right)}$

4. The product rule

$x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}$
$x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}$

$\left(x + 4\right) e^{x}$
$\left(x + 9\right) e^{x}$

$-20$
$-16$
$6$

$x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)}$
$x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}$
$x^{5} \cos{\left(x \right)} + 5 x^{4} \sin{\left(x \right)}$

5. The quotient rule

$\frac{6}{\left(x + 6\right)^{2}}$
$\frac{3}{\left(x + 3\right)^{2}}$
$\frac{2}{\left(x + 2\right)^{2}}$

$\frac{4 x \cos{\left(x \right)} - 4 \sin{\left(x \right)}}{x^{2}}$
$\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}}$

$- \frac{5}{8}$
$-3$

$\frac{6}{\left(x + 6\right)^{2}}$
$\frac{3}{\left(x + 3\right)^{2}}$
$\frac{2}{\left(x + 2\right)^{2}}$

6. Derivatives of tangent, secant, cotangent and cosecant

$7 \sec^{2}{\left(x \right)}$
$3 \sec^{2}{\left(x \right)}$
$5 \sec^{2}{\left(x \right)}$

$\tan{\left(x \right)} \sec{\left(x \right)} - 2 \csc^{2}{\left(x \right)}$
$\tan{\left(x \right)} \sec{\left(x \right)} - 8 \csc^{2}{\left(x \right)}$

$\frac{4}{3}$
$2$

$7 \sec^{2}{\left(x \right)}$
$3 \sec^{2}{\left(x \right)}$
$5 \sec^{2}{\left(x \right)}$

7. The chain rule

$24 \left(6 x + 7\right)^{3}$
$28 \left(7 x + 6\right)^{3}$
$12 \left(3 x + 5\right)^{3}$

$8 x \cos{\left(4 x^{2} \right)}$
$12 x \cos{\left(6 x^{2} \right)}$

$10 x e^{5 x^{2}}$
$8 x e^{4 x^{2}}$

$- \frac{\pi^{3}}{16}$

$24 \left(6 x + 7\right)^{3}$
$28 \left(7 x + 6\right)^{3}$
$12 \left(3 x + 5\right)^{3}$

8. Derivatives of inverse functions: ln x, arcsin x, arctan x

$\frac{1}{x}$
$\frac{1}{x}$

$\frac{2 x}{x^{2} + 2}$
$\frac{2 x}{x^{2} + 8}$

$\frac{7}{49 x^{2} + 1}$
$\frac{2}{4 x^{2} + 1}$

$\frac{2 x}{x^{2} + 2}$
$\frac{2 x}{x^{2} + 8}$
$\frac{2 x}{x^{2} + 1}$