Edexcel IAL Mathematics FP3.3.2 inverse function differentiation

Inverse-function differentiation questions reward clean chain-rule work: keep the domain restriction, simplify radicals or quotients and solve exact f′(x)=0 equations.

[Maximum number: 7]

$\quad y=\operatorname{arsinh}\left(\sqrt{x^{2}-1}\right) \quad x>1$

(a)

Prove that

\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\sqrt{x^2-1}}

where

y=\operatorname{arsinh}\left(\sqrt{x^2-1}\right),\quad x>1.

[ 3 ]

\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\sqrt{1+\left(\sqrt{x^2-1}\right)^2}}\cdot \frac{x}{\sqrt{x^2-1}}

=\frac{1}{\sqrt{x^2}}\cdot\frac{x}{\sqrt{x^2-1}}

Since x>1, $\sqrt{x^2}=x$ , so

\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\sqrt{x^2-1}}

M1: differentiates arsinh(sqrt(x^2 - 1)) using the chain rule.
A1: fully correct unsimplified derivative.
A1*: correct completion to the required result with no errors.
Other official methods using hyperbolic identities or logarithmic form are accepted.

(b)

Determine the exact values of x for which f'(x)=0.

[ 4 ]

f'(x)=\frac{1}{3\sqrt{x^2-1}}-\frac{1}{1+x^2}

f'(x)=0\Rightarrow 1+x^2=3\sqrt{x^2-1}

(1+x^2)^2=9(x^2-1)

x^4-7x^2+10=0

(x^2-2)(x^2-5)=0

x=\sqrt{2},\ \sqrt{5}

M1: forms a correct derivative equation with denominator 1 + x^2 for arctan x.
M1: cross-multiplies and squares correctly.
ddM1: solves the three-term quadratic in x^2.
A1: both exact positive solutions only.

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FP3.3.2 - Differentiation of inverse functions, question 1

Question (a)

Question (b)