A-level · Differentiation

Differentiating exponentials and logarithmsExam-style questions

The functions whose rate of change is proportional to their size. Why the gradient of any exponential is its height times a constant; the base e, which makes that constant exactly 1; e^(kx) with the chain rule; other bases and their factor of ln a; the derivative of ln x and of ln of anything; and exponentials and logs with the product and quotient rules.

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19 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Differentiate $y = 2^x$ , giving your answer exactly.
Level 3 · 2 marksTry it
2
Show that the curve $y = x\ln x - x$ has gradient $\ln x$ , and hence find the coordinates of its stationary point.
Level 3 · 4 marksTry it
3
Find an equation of the tangent to the curve $y = \ln x$ at the point where $x = \mathrm{e}$ . (3 marks)
Level 3 · 3 marksTry it
4
(a) Differentiate $y = 3^{x}$ , giving your answer exactly. (2 marks)
(b) Hence find the exact gradient of the curve at $x = 2$ . (1 mark)
Level 3 · 3 marksTry it
5
Differentiate $y = \dfrac{\ln x}{x}$ , and hence find the $x$ -coordinate of the stationary point.
Level 3 · 3 marksTry it

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