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Paper 1 · November 2010 · Integration and Reduction Formulae

Given that $y = \sinh^{-1} x$ , find $\frac{dy}{dx}$ .

Model answer

1/sqrt(x^2+1)

Also accepted: 1/sqrt(1+x^2), (x^2+1)^(-1/2)

Explanation

From

y=\sinh^{-1}x

,

x=\sinh y

, so

\dfrac{dx}{dy}=\cosh y

. Since

\cosh^2y-\sinh^2y=1

and

\cosh y

is always positive,

\cosh y=\sqrt{1+\sinh^2 y}=\sqrt{1+x^2}

. Hence

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

.

Derived from ZIMSEC Further Mathematics 9187/1, November 2010, Q13

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