Pure Maths

A-Level Maths | Differentiation

Differentiating Exponential, Logarithmic and Trigonometric Functions

$$e^{kx} \rightarrow ke^{kx}$$ $$\ln(x) \rightarrow \dfrac{1}{x} \qquad \ln(kx) \rightarrow \dfrac{k}{kx} \rightarrow \dfrac{1}{x} \qquad \ln(kx+c) \rightarrow \dfrac{k}{kx+c}$$ $$\sin(kx) \rightarrow k\cos(kx)$$ $$\cos(kx) \rightarrow -k\sin(kx)$$ $$a^x \rightarrow a^x\ln(a)$$

Differentiating exponential functions

Question 1

Differentiate with respect to $x$.

a) $e^{3x}$

b) $e^{10x}$

c) $e^{14x}$

d) $e^{-3x}$

e) $e^{-12x}$

f) $e^{2x}$

g) $e^{-x}$

Show Answers

a) $3e^{3x}$

b) $10e^{10x}$

c) $14e^{14x}$

d) $-3e^{-3x}$

e) $-12e^{-12x}$

f) $2e^{2x}$

g) $-e^{-x}$

Differentiating logarithmic functions

Question 2

Differentiate with respect to $x$.

a) $\ln(x)$

b) $\ln(2x)$

c) $\ln(5x)$

d) $\ln(14x)$

e) $\ln(x+1)$

f) $\ln(x+3)$

g) $\ln(x+5)$

h) $\ln(x+7)$

i) $\ln(x+10)$

j) $\ln(2x+4)$

k) $\ln(5x+10)$

l) $\ln(3x+1)$

m) $\ln(4x+7)$

n) $\ln(7x+2)$

Show Answers

a) $\dfrac{1}{x}$

b) $\dfrac{1}{x}$

c) $\dfrac{1}{x}$

d) $\dfrac{1}{x}$

e) $\dfrac{1}{x+1}$

f) $\dfrac{1}{x+3}$

g) $\dfrac{1}{x+5}$

h) $\dfrac{1}{x+7}$

i) $\dfrac{1}{x+10}$

j) $\dfrac{2}{2x+4}$

k) $\dfrac{5}{5x+10}$

l) $\dfrac{3}{3x+1}$

m) $\dfrac{4}{4x+7}$

n) $\dfrac{7}{7x+2}$

Differentiating trigonometric functions

Question 3

Differentiate with respect to $x$.

a) $\sin(2x)$

b) $\sin(4x)$

c) $\sin(-12x)$

d) $\sin(5x)$

e) $\sin(-3x)$

f) $\cos(2x)$

g) $\cos(4x)$

h) $\cos(3x)$

i) $\cos(5x)$

j) $\cos(-4x)$

Show Answers

a) $2\cos(2x)$

b) $4\cos(4x)$

c) $-12\cos(-12x)$

d) $5\cos(5x)$

e) $-3\cos(-3x)$

f) $-2\sin(2x)$

g) $-4\sin(4x)$

h) $-3\sin(3x)$

i) $-5\sin(5x)$

j) $4\sin(-4x)$

The Chain Rule

Question 4

Differentiate with respect to $x$.

a) $y=(2x+1)^4$

b) $y=(3x-2)^6$

c) $y=(8x-1)^{-1}$

d) $y=(6x+1)^{\frac{1}{2}}$

e) $y=(4x+3)^{\frac{3}{2}}$

f) $y=(1-x)^4$

g) $y=2(5x-2)^3$

h) $y=2(3x+1)^{\frac{1}{2}}$

i) $y=4(1-5x)^{-\frac{1}{2}}$

j) $y=6(1-2x)^{\frac{1}{3}}$

Show Answers

a) $8(2x+1)^3$

b) $18(3x-2)^5$

c) $\dfrac{-8}{(8x-1)^2}$

d) $\dfrac{3}{\sqrt{6x+1}}$

e) $6\sqrt{4x+3}$

f) $-4(1-x)^3$

g) $30(5x-2)^2$

h) $\dfrac{3}{\sqrt{3x+1}}$

i) $\dfrac{10}{(1-5x)^{\frac{3}{2}}}$

j) $\dfrac{-4}{(1-2x)^{\frac{2}{3}}}$

Question 5

a) $y=\dfrac{3}{\sqrt{3-2x}}$

b) $y=\dfrac{1}{4x+1}$

c) $y=\dfrac{2}{3(2x+7)^2}$

d) $y=\dfrac{3}{x^2+1}$

e) $y=\dfrac{4}{\sqrt{4-3x^2}}$

f) $y=\dfrac{1}{1-x^3}$

Show Answers

a) $\dfrac{3}{(3-2x)^{\frac{3}{2}}}$

b) $\dfrac{-4}{(4x+1)^2}$

c) $\dfrac{-8}{3(2x+7)^3}$

d) $\dfrac{-6x}{(x^2+1)^2}$

e) $\dfrac{12x}{(4-3x^2)^{\frac{3}{2}}}$

f) $\dfrac{3x^2}{(1-x^3)^2}$

Chain Rule with Exponentials

Question 6

Differentiate with respect to $x$.

a) $y=3e^{x^2}$

b) $y=e^{3x+1}$

c) $y=2e^{x^3}$

d) $y=(1+e^x)^4$

e) $y=e^{\sin x}$

f) $y=4e^{2x^2}$

g) $y=e^{-x^2}$

Show Answers

a) $6xe^{x^2}$

b) $3e^{3x+1}$

c) $6x^2e^{x^3}$

d) $4e^x(1+e^x)^3$

e) $e^{\sin x}\cos x$

f) $16xe^{2x^2}$

g) $-2xe^{-x^2}$

Chain Rule with Logarithms

Question 7

Differentiate with respect to $x$.

a) $y=\ln(x^2-4)$

b) $y=\ln(x^3+1)$

c) $y=\ln(x^2+x)$

d) $y=(\ln x)^4$

e) $y=(\ln x)^3$

f) $y=\ln x^4$

g) $y=\ln(\sin x)$

Show Answers

a) $\dfrac{2x}{x^2-4}$

b) $\dfrac{3x^2}{x^3+1}$

c) $\dfrac{2x+1}{x^2+x}$

d) $\dfrac{4(\ln x)^3}{x}$

e) $\dfrac{3(\ln x)^2}{x}$

f) $\dfrac{4}{x}$

g) $\dfrac{\cos x}{\sin x}$

Chain Rule with Trig

Question 8

Differentiate with respect to $x$.

a) $y=2\sin(x^4)$

b) $y=2\sin^4 x$

c) $y=4\cos(x^3)$

d) $y=4\cos^3 x$

e) $y=3\sin^5 x$

f) $y=2\cos\sqrt{x}$

g) $y=2\sin^3 2x$

Show Answers

a) $8x^3\cos(x^4)$

b) $8\sin^3 x\cos x$

c) $-12x^2\sin(x^3)$

d) $-12\sin x\cos^2 x$

e) $15\sin^4 x\cos x$

f) $-\dfrac{\sin\sqrt{x}}{\sqrt{x}}$

g) $12\sin^2(2x)\cos(2x)$

Product Rule

$$\frac{d}{dx}(uv) = uv' + vu'$$

Question 9

Differentiate with respect to $x$.

a) $x(1+3x)^5$

b) $2x(1+3x^2)^3$

c) $x^3(2x+6)^4$

d) $3x^2(5x-1)^{-1}$

Show Answers

a) $(3x+1)^4(18x+1)$

b) $2(3x^2+1)^2(21x^2+1)$

c) $16x^2(x+3)^3(7x+9)$

d) $3x(5x-2)(5x-1)^{-2}$

Quotient Rule

$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{vu' - uv'}{v^2}$$

Question 10

Differentiate with respect to $x$.

a) $y=\dfrac{5x}{x+1}$

b) $y=\dfrac{2x}{3x-2}$

c) $y=\dfrac{x+3}{2x+1}$

d) $y=\dfrac{3x^2}{(2x-1)^2}$

e) $y=\dfrac{6x}{(5x+3)^{\frac{1}{2}}}$

Show Answers

a) $\dfrac{5}{(x+1)^2}$

b) $-\dfrac{4}{(3x-2)^2}$

c) $-\dfrac{5}{(2x+1)^2}$

d) $-\dfrac{6x}{(2x-1)^3}$

e) $\dfrac{15x+18}{(5x+3)^{\frac{1}{2}\cdot 3}}$

Factorising Difficult Expressions

Question 11

Factorise each of the following expressions fully.

a) $x \times (2x+1)^{-\frac{1}{2}} + \sqrt{2x+1} \times 1$

b) $x^2 \times 2(4x+1)^{-\frac{1}{2}} + \sqrt{4x+1} \times 2x$

c) $x^3 \times \dfrac{5}{2}(5x-2)^{-\frac{1}{2}} + \sqrt{5x-2} \times 3x^2$

d) $x \times 6(2x+1)^2 + (2x+1)^3 \times 1$

e) $x^2 \times 4(x+3)^3 + (x+3)^4 \times 2x$

f) $x^3 \times (-6)(3x+1)^{-3} + (3x+1)^{-2} \times 3x^2$

g) $x^2 \times \left(-\dfrac{3}{2}\right)(1-x)^{\frac{1}{2}} + (1-x)^{\frac{3}{2}} \times 2x$

h) $(x+3)^2 \times (2x-1)^{-\frac{1}{2}} + \sqrt{2x-1} \times 2(x+3)$

i) $x^2 \times 3e^{3x} + e^{3x} \times 2x$

j) $x \times x(x^2+4)^{-\frac{1}{2}} + \sqrt{x^2+4} \times 1$

Show Answers

a) $\dfrac{3x+1}{\sqrt{2x+1}}$

b) $\dfrac{2x(5x+1)}{\sqrt{4x+1}}$

c) $\dfrac{x^2(35x-12)}{2\sqrt{5x-2}}$

d) $(2x+1)^2(8x+1)$

e) $6x(x+1)(x+3)^3$

f) $\dfrac{3x^2(x+1)}{(3x+1)^3}$

g) $\dfrac{x\sqrt{1-x}\,(4-7x)}{2}$

h) $\dfrac{(x+3)(5x+1)}{\sqrt{2x-1}}$

i) $xe^{3x}(3x+2)$

j) $\dfrac{2(x^2+2)}{\sqrt{x^2+4}}$

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