9.2PureStretch

Differentiating Exponentials and Logarithms

The exponential function eˣ is its own derivative — a defining property. From it we get the derivatives of general exponentials and of the natural logarithm.

26 min Differentiation

Edexcel A level Maths: 9.2 Differentiating Exponential and Logarithmic FunctionsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Differentiate eˣ and eᵏˣ
Differentiate aˣ
Differentiate ln x
Combine with the chain rule

The exponential

$\dfrac{d}{dx}(e^x) = e^x$ . With the chain rule, $\dfrac{d}{dx}(e^{kx}) = k e^{kx}$ . For a general base, $\dfrac{d}{dx}(a^x) = a^x \ln a$ .

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

eˣ is its own derivative.

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

General base.

The natural logarithm

$\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}$ . This is one of the most useful derivatives in integration too.

Example 1Differentiate

y = e^{3x}

\tfrac{dy}{dx} = 3e^{3x}

(chain rule, k = 3).

Answer

3e^{3x}

Tip — d/dx(ln x) = 1/x is only defined for x > 0.

Formula recap

\frac{d}{dx}(e^{kx}) = k e^{kx}

Exponential.

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

General base.

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

Natural log.

Common mistakes to avoid

Differentiating eˣ to x eˣ⁻¹.

eˣ differentiates to itself, eˣ.

Forgetting the ln a factor for aˣ.

d/dx(aˣ) = aˣ ln a.

Key takeaways

d/dx(eˣ) = eˣ; d/dx(eᵏˣ) = k eᵏˣ.
d/dx(aˣ) = aˣ ln a.
d/dx(ln x) = 1/x.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

$\frac{d}{dx}(e^x)$ equals:
- $e^x$
- $x e^{x-1}$
- $e^{x-1}$
- $\ln x$
Show answer and explanation
Answer: $e^x$
Its own derivative.
$\frac{d}{dx}(e^{3x})$ equals:
- $3e^{3x}$
- $e^{3x}$
- $3x e^{3x}$
- $e^{3}$
Show answer and explanation
Answer: $3e^{3x}$
Chain rule, k = 3.
$\frac{d}{dx}(\ln x)$ equals:
- $\frac{1}{x}$
- $x$
- $\ln x$
- $e^x$
Show answer and explanation
Answer: $\frac{1}{x}$
Standard result.
$\frac{d}{dx}(a^x)$ equals:
- $a^x \ln a$
- $x a^{x-1}$
- $a^x$
- $a \ln x$
Show answer and explanation
Answer: $a^x \ln a$
General base.
$\frac{d}{dx}(\ln x)$ is defined for:
- $x > 0$
- all x
- $x \ne 0$
- $x < 0$
Show answer and explanation
Answer: $x > 0$
Domain of ln.

Next step

Check your understanding

Work through questions on Differentiating Exponentials and Logarithms. Use the feedback to decide what needs another look.

3 past-paper questions · 9 marks

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