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
The curve $C$ has equation $y = 2^{x^2 - 6x}$ .
(a) Determine the coordinates of each point on $C$ at which the gradient is zero.

(4 marks)
(b) Show that $C$ is convex for every value of $x$ .

(5 marks)
Level 4 · 9 marksTry it
2
$\text{f}(x) = e^{x^3 + 1} \qquad x > 0$

$\text{g}(x) = 3\ln x \qquad x > 0$
(a) Find

(i) $\text{f}'(x)$

(ii) $\text{g}'(x)$

(2 marks)

There is exactly one value of $x$ , $\alpha$ , for which $\text{f}'(x) = \text{g}'(x)$
(b) Show that the equation $\text{f}'(x) = \text{g}'(x)$ can be written as

$x^3 + 3\ln x + 1 = 0$

(2 marks)

To find an approximation for $\alpha$ , the iterative formula

$x_{n+1} = e^{-\frac{1}{3}\left(x_n^3 + 1\right)}$

is applied, starting with $x_1 = 0.7$
(c) Calculate, to 4 decimal places,

(i) the value of $x_2$

(ii) the value of $\alpha$

(3 marks)
Level 4 · 7 marksTry it
3
The curve $C$ has equation $y = \mathrm{e}^{3x} - 3x$ .
(a) Find the coordinates of the stationary point of $C$ . (3 marks)
(b) Determine its nature. (2 marks)
Level 4 · 5 marksTry it