A-level · Differentiation

Differentiating powers of xExam-style questions

Finding how steep a curve is at any point. What a derivative measures and where it comes from; the power rule and why the power comes down; numbers in front, sums and constants; rewriting roots and fractions as powers; negative and fractional powers; using the gradient function at a point and backwards; and differentiating twice.

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16 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
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm{f}(x)$ , where

$\mathrm{f}(x) = (x + 3)^2(5 - x)$
(a) Using Figure 1, deduce the set of values of $x$ for which $\mathrm{f}(x) \geqslant 0$

(1 mark)
(b) Find $\mathrm{f}'(x)$ , giving your answer in simplest form.

(3 marks)

The point $A(1,\ 64)$ lies on $C$ . The tangent to $C$ at $A$ and the tangent to $C$ at a different point $B$ have the same gradient.
(c) (i) Show that the $x$ coordinate of $B$ satisfies the equation

$3x^2 + 2x - 5 = 0$

(ii) Hence find the $x$ coordinate of $B$ .

(4 marks)
(d) State the coordinates of the image of $A$ on the curve with equation

(i) $y = \mathrm{f}(-x)$

(ii) $y = \tfrac12\mathrm{f}(x)$

(2 marks)
Level 5 · 10 marksTry it