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 Video by Zeeshan Zamurred Differentiation

Edexcel A level Maths: 9.2 Differentiating Exponential and Logarithmic FunctionsWatch the full walkthrough 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

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

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

eˣ is its own derivative.

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

General base.

The natural logarithm

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

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

(chain rule, k = 3).

Answer

3 e^{3 x}

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

Formula recap

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

Exponential.

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

General base.

\frac{d}{d x} (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.

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