Differentiating Exponentials, Trigonometric and Logarithmic Functions — Edexcel A-level Mathematics (Pure)

SPEC 7.2 • DIFFERENTIATION

Differentiating Exponentials, Trig & Logs

Standard results for eˣ, sin x, cos x, tan x and ln x

Exponential Functions

The exponential function

e^{x}

has a special property: it differentiates to itself.

Standard Results

\frac{d}{d x} (e^{x}) = e^{x}

\frac{d}{d x} (e^{k x}) = k e^{k x}

General Exponential

For any base

a

:

\frac{d}{d x} (a^{k x}) = k a^{k x} ln a

Remember

For

e^{k x}

, the coefficient

k

comes down as a multiplier. The power stays exactly the same!

01Worked Example

Differentiate:

(a)

y = e^{3 x}

(b)

f (x) = 5 e^{2 x}

(c)

y = 4 e^{- x} + e^{3 x}

(d)

y = 2^{3 x}

•

(a)

y = e^{3 x}

Using

\frac{d}{d x} (e^{k x}) = k e^{k x}

with

k = 3

:

\frac{d y}{d x} = 3 e^{3 x}

•

(b)

f (x) = 5 e^{2 x}

The 5 multiplies the derivative:

f^{'} (x) = 5 \times 2 e^{2 x} = 10 e^{2 x}

•

(c)

y = 4 e^{- x} + e^{3 x}

Differentiate term by term (note

k = - 1

for first term):

\frac{d y}{d x} = 4 \times (- 1) e^{- x} + 3 e^{3 x} = - 4 e^{- x} + 3 e^{3 x}

•

(d)

y = 2^{3 x}

Using

\frac{d}{d x} (a^{k x}) = k a^{k x} ln a

with

a = 2, k = 3

:

\frac{d y}{d x} = 3 \times 2^{3 x} \times ln 2 = 3 \cdot 2^{3 x} ln 2

Trigonometric Functions

The derivatives of the basic trig functions must be memorised:

Standard Results

\frac{d}{d x} (sin x) = cos x

\frac{d}{d x} (cos x) = - sin x

\frac{d}{d x} (tan x) = sec^{2} x

Critical

The derivative of

cos x

is

- sin x

(negative!). This is the most common error.

Memory Aid

Think "sin → cos → -sin → -cos → sin" as you go around the cycle of derivatives.

Trig with Multiples (kx)

When the argument is

k x

instead of

x

, the coefficient

k

comes out:

Extended Results

\frac{d}{d x} (sin k x) = k cos k x

\frac{d}{d x} (cos k x) = - k sin k x

\frac{d}{d x} (tan k x) = k sec^{2} k x

Pattern

The

k

multiplies the entire derivative, and the argument stays as

k x

throughout.

02Worked Example

Differentiate:

(a)

y = sin 3 x

(b)

f (x) = 4 cos 2 x

(c)

y = 2 tan 5 x

•

(a)

y = sin 3 x

Using

\frac{d}{d x} (sin k x) = k cos k x

with

k = 3

:

\frac{d y}{d x} = 3 cos 3 x

•

(b)

f (x) = 4 cos 2 x

Using

\frac{d}{d x} (cos k x) = - k sin k x

:

f^{'} (x) = 4 \times (- 2 sin 2 x) = - 8 sin 2 x

•

(c)

y = 2 tan 5 x

Using

\frac{d}{d x} (tan k x) = k sec^{2} k x

:

\frac{d y}{d x} = 2 \times 5 sec^{2} 5 x = 10 sec^{2} 5 x

Logarithmic Functions

The natural logarithm

ln x

has a simple derivative:

Standard Results

\frac{d}{d x} (ln x) = \frac{1}{x}

\frac{d}{d x} (ln k x) = \frac{1}{x}

Why $ln k x$ gives $\frac{1}{x}$

Using log laws:

ln k x = ln k + ln x

Since

ln k

is a constant, its derivative is 0, leaving just

\frac{1}{x}

.

Note

The domain of

ln x

requires

x > 0

. The derivative

\frac{1}{x}

is only valid for positive

x

.

03Worked Example

Differentiate:

(a)

y = ln x

(b)

f (x) = 5 ln x

(c)

y = ln 3 x

(d)

f (x) = 4 ln 2 x

•

(a)

y = ln x

\frac{d y}{d x} = \frac{1}{x}

•

(b)

f (x) = 5 ln x

f^{'} (x) = 5 \times \frac{1}{x} = \frac{5}{x}

•

(c)

y = ln 3 x

The constant inside disappears:

\frac{d y}{d x} = \frac{1}{x}

•

(d)

f (x) = 4 ln 2 x

The 4 multiplies, the 2 inside disappears:

f^{'} (x) = 4 \times \frac{1}{x} = \frac{4}{x}

04Worked Example

Differentiate:

(a)

y = e^{2 x} + sin 3 x

(b)

f (x) = 2 cos x - 3 ln x

•

(a)

y = e^{2 x} + sin 3 x

Differentiate each term separately:

\frac{d y}{d x} = 2 e^{2 x} + 3 cos 3 x

•

(b)

f (x) = 2 cos x - 3 ln x

Remember:

\frac{d}{d x} (cos x) = - sin x

f^{'} (x) = 2 (- sin x) - 3 \times \frac{1}{x} = - 2 sin x - \frac{3}{x}

05Worked Example

Differentiate:

(a)

y = x^{2} + e^{x} - cos 2 x

(b)

f (x) = ln x + tan x - e^{- x}

•

(a)

y = x^{2} + e^{x} - cos 2 x

Apply each rule:

\frac{d}{d x} (x^{2}) = 2 x

\frac{d}{d x} (e^{x}) = e^{x}

\frac{d}{d x} (- cos 2 x) = - (- 2 sin 2 x) = 2 sin 2 x

\frac{d y}{d x} = 2 x + e^{x} + 2 sin 2 x

•

(b)

f (x) = ln x + tan x - e^{- x}

Apply each rule:

\frac{d}{d x} (ln x) = \frac{1}{x}

\frac{d}{d x} (tan x) = sec^{2} x

\frac{d}{d x} (- e^{- x}) = - (- 1) e^{- x} = e^{- x}

f^{'} (x) = \frac{1}{x} + sec^{2} x + e^{- x}

Practice

Differentiate:

(a)

y = e^{4 x}

(b)

f (x) = 3 sin 2 x

(c)

y = 5 cos 3 x

•

(a)

\frac{d y}{d x} = 4 e^{4 x}

•

(b)

f^{'} (x) = 3 \times 2 cos 2 x = 6 cos 2 x

•

(c)

\frac{d y}{d x} = 5 \times (- 3 sin 3 x) = - 15 sin 3 x

Practice

Differentiate:

(a)

y = 6 ln x

(b)

f (x) = tan 4 x

(c)

y = e^{- 2 x} + 3 ln x

•

(a)

\frac{d y}{d x} = \frac{6}{x}

•

(b)

f^{'} (x) = 4 sec^{2} 4 x

•

(c)

\frac{d y}{d x} = - 2 e^{- 2 x} + \frac{3}{x}

Practice

Differentiate:

(a)

y = x^{3} - 2 e^{x} + sin x

(b)

f (x) = 4 cos 2 x + e^{3 x} - ln x

•

(a)

\frac{d y}{d x} = 3 x^{2} - 2 e^{x} + cos x

•

(b)

f^{'} (x) = 4 (- 2 sin 2 x) + 3 e^{3 x} - \frac{1}{x} = - 8 sin 2 x + 3 e^{3 x} - \frac{1}{x}

Summary

All Standard Results

FunctionDerivative

e^{x}

e^{x}

e^{k x}

k e^{k x}

a^{k x}

k a^{k x} ln a

sin k x

k cos k x

cos k x

- k sin k x

tan k x

k sec^{2} k x

ln x

\frac{1}{x}

Exam Tips

Memorise all results - they appear on the formula sheet but you need them quickly

Watch the negative sign when differentiating cos

For mixed expressions, differentiate each term separately