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Differentiation — WFM03 IAL Further Mathematics practice questions · Study-IAL

Subjects
Further Mathematics
WFM03
3.3

Differentiation

WFM03Topic 3.33 questions

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Differentiating hyperbolic and inverse trigonometric and hyperbolic functions.

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Questions

Multiple choice: 1
Short answer: 2
Total: 3

Previous topic3.2Further coordinate systems Next topic 3.4Integration

Sample questions

Original practice questions written in the style of the exam.

Question 14 marks
Given that $y = \arctan(\sinh x)$ , show that $\dfrac{dy}{dx} = \operatorname{sech} x$ .
Mark scheme
- M1 Uses $\dfrac{d}{du}\arctan u = \dfrac{1}{1 + u^2}$ with the chain rule (1)
Question 24 marks
Given that $y = \operatorname{arsinh} x$ , show that $(1 + x^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx} = 0$

u2

1

A1

\dfrac{dy}{dx} = \dfrac{\cosh x}{1 + \sinh^2 x}

(1)

M1 Uses

1 + \sinh^2 x = \cosh^2 x

(1)

A1

\dfrac{\cosh x}{\cosh^2 x} = \dfrac{1}{\cosh x} = \operatorname{sech} x

(1)

2

)

dx2d2y

+

xdxdy=

0

Mark scheme

B1 $\dfrac{dy}{dx} = (1 + x^2)^{-1/2}$ (1)
M1 Differentiates again: $\dfrac{d^2y}{dx^2} = -\tfrac{1}{2}(1 + x^2)^{-3/2} \cdot 2x$ (1)
A1 $\dfrac{d^2y}{dx^2} = -x(1 + x^2)^{-3/2}$ (1)
A1 $(1 + x^2)\left(-x(1 + x^2)^{-3/2}\right) + x(1 + x^2)^{-1/2} = -x(1 + x^2)^{-1/2} + x(1 + x^2)^{-1/2} = 0$ (1)

=

−21(1+

x2)−3/2⋅

2x

−

x

(

1

+

x2)−3/2

(

−

x

(

1

+

x2

)−3/2

)

+

x(1+

x2)−1/2=

−x(1+

x2)−1/2+

x(1+

x2)−1/2=

0