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# A-Level Maths | Differentiation
- URL: https://www.a-level-maths-tutor.com/a-level-maths-differentiation/
- Published: 2026-09-23T11:03:18.000Z
- Updated: 2026-10-06T09:35:07.000Z
- Author: Jaisul Naik
- Tags: Pure Maths

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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