Pearson Edexcel IAL Mathematics P3.4.2 Differentiation

Practise product, quotient and chain rule differentiation for functions, curves and models, with derivatives simplified into required forms.

Exam points

apply product, quotient or chain rules to functions involving fractions, powers, trig and e^x
use f'(x) to prove a function is increasing or to form a stationary-point equation
find tangent or normal gradients from a derivative and connect them to curve information

[Maximum number: 3]

A scientist is studying a population of fish in a lake. The number of fish, N, in the population, t years after the start of the study, is modelled by the equation

N=\frac{600 \mathrm{e}^{0.3 t}}{2+\mathrm{e}^{0.3 t}} \quad t \geqslant 0

Use the equation of the model to answer parts (a), (b), (c), (d) and (e).

Show that

$\frac{dN}{dt}=\frac{Ae^{0.3t}}{(2+e^{0.3t})^2}$

where A is a constant to be found.

$N=\frac{600e^{0.3t}}{2+e^{0.3t}}$ $\frac{dN}{dt}=\frac{(2+e^{0.3t})(180e^{0.3t})-600e^{0.3t}(0.3e^{0.3t})}{(2+e^{0.3t})^2}$
M1 A1
$\frac{dN}{dt}=\frac{360e^{0.3t}}{(2+e^{0.3t})^2}$
A1

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P3.4.2 - Differentiation question 1