Edexcel A-Level Mathematics AS P1.4 Differentiation Questions

Practise differentiating functions, interpreting gradients and forming tangents, normals and rates of change from algebraic models.

Exam points

differentiate powers of x, roots and reciprocal terms before simplifying dy/dx
use dy/dx at a point to form tangent or normal equations in y=mx+c form

Covered syllabus points

P1.4.1 - Derivative as gradient and rate of change P1.4.2 - Differentiating powers of x P1.4.3 - Tangents, normals and gradients

[Maximum number: 2]

A curve C has equation y=f(x) where

f(x)=(2-k x)^{5}

and k is a constant.
Given that when f(x) is divided by (4 x-5) the remainder is $\frac{243}{32}$

find the gradient of C at the point where x=0

$f'(x)=-32+\frac{128}{5}x$ f'(0)=-32.
M1 A1ft

[Maximum number: 3]

\mathrm{f}(x)=10-2 x-\frac{1}{2 \sqrt{x}}-\frac{1}{x^{3}} \quad x>0

Determine $\mathrm{f}^{\prime}(x)$ .

$f^{\prime}(x)=-2+\frac{1}{4} x^{-\frac{3}{2}}+3 x^{-4}$ }

$x^{n} \rightarrow x^{n-1}$ in at least 1 term other than
10
2 of the 3 terms shown correct
All correct
(3)

[Maximum number: 4]

The curve C has equation y=f(x), x>0 where

\mathrm{f}^{\prime}(x)=30+\frac{6-5 x^{2}}{\sqrt{x}}

Given that the point P(4,-8) lies on C,

find the equation of the normal to C at P, giving your answer in the form y=m x+c, where m and c are constants.

f'(4)=30+\frac{6-5(4)^2}{\sqrt4}=-7

y+8=\frac17(x-4)

y=\frac17x-\frac{60}{7}

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Covered syllabus points

Question 1

Question 2

Question 3