CAIE A-Level Mathematics 2.4.1 Advanced Differentiation

Practise differentiating eˣ, ln x and trigonometric composites with product, quotient and chain rules and solving derivative conditions for exact gradients or stationary points.

Exam points

differentiate eᵘ, ln u or a trig function and multiply by the derivative of u
combine product or quotient terms before simplifying the derivative into a solvable form
set the derivative to zero or evaluate it at the stated point to obtain an exact result

[Maximum number: 3]

Figure for Question 2.4.1—Advanced differentiation question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}$ . The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

Find the gradient of the curve at A.

Differentiate to obtain form $k_{1} \mathrm{e}^{-x}+k_{2} \mathrm{e}^{2 x}$

M1

Where $k_{1} k_{2} \neq 0, k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $-8 \mathrm{e}^{-x}-2 \mathrm{e}^{2 x}$

A1

Substitute x=0 to obtain -10

A1

All question bank results loaded

2.4.1—Advanced differentiation question 1