Pearson Edexcel IAL Mathematics P3.4.1 Differentiating exp, log & trig functions

Practise differentiating e^kx, ln(kx) and trig functions to find gradients, rates of change and simplified derivative forms in context.

Exam points

find dy/dx for sin(kx), cos(kx) or e^kt, then evaluate a tangent gradient or rate
use dx/dy and trig identities to rewrite dy/dx in a required form involving x
differentiate model formulae and use the derivative to form an iteration or solve for a time

[Maximum number: 5]

A curve C has equation

y=x^{\sin x} \quad x>0 \quad y>0

Find, by firstly taking natural logarithms, an expression for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of x and y.

$y=x^{\sin x}\Rightarrow \ln y=\sin x\ln x$

B1

$\frac1y\frac{dy}{dx}=\frac{\sin x}{x}+\ln x\cos x$

M1 M1 A1

$\frac{dy}{dx}=y\left(\frac{\sin x}{x}+\ln x\cos x\right)$

A1

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P3.4.1 - Differentiating exp, log and trig functions question 1