Inverse functions (inverse trig/hyperbolic)

Questions where the base function is an inverse trigonometric or inverse hyperbolic function (e.g., tan⁻¹(2x), sin⁻¹(2x), sech⁻¹(x+1/2), tanh⁻¹(x)), requiring implicit differentiation or chain rule with inverse function derivatives.

5 questions

CAIE Further Paper 2 2021 June Q7

It is given that $\mathrm { y } = \operatorname { sech } ^ { - 1 } \left( \mathrm { x } + \frac { 1 } { 2 } \right)$.
Express cosh $y$ in terms of $x$ and hence show that $\sinh y \frac { d y } { d x } = - \frac { 1 } { \left( x + \frac { 1 } { 2 } \right) ^ { 2 } }$.
Find the first three terms in the Maclaurin's series for $\operatorname { sech } ^ { - 1 } \left( x + \frac { 1 } { 2 } \right)$ in the form $$\ln a + b x + c x ^ { 2 }$$ where $a$, $b$ and $c$ are constants to be determined.

OCR FP2 2016 June Q5

5 It is given that $y = \tan ^ { - 1 } 2 x$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 0$.
Find the Maclaurin series for $y$ up to and including the term in $x ^ { 3 }$. Show all your working.
The result in part (ii), together with the value $x = \frac { 1 } { 2 }$, is used to find an estimate for $\pi$. Show that this estimate is only correct to 1 significant figure.

OCR FP2 2015 June Q5

5 It is given that $y = \sin ^ { - 1 } 2 x$.

Using the derivative of $\sin ^ { - 1 } x$ given in the List of Formulae (MF1), find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Show that $\left( 1 - 4 x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x }$.
Hence show that $\left( 1 - 4 x ^ { 2 } \right) \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } - 12 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0$.
Using your results from parts (i), (ii) and (iii), find the Maclaurin series for $\sin ^ { - 1 } 2 x$ up to and including the term in $x ^ { 3 }$.

Edexcel CP2 2024 June Q2

2. $$f ( x ) = \tanh ^ { - 1 } \left( \frac { 3 - x } { 6 + x } \right) \quad | x | < \frac { 3 } { 2 }$$

Show that $$f ^ { \prime } ( x ) = - \frac { 1 } { 2 x + 3 }$$
Hence determine $\mathrm { f } ^ { \prime \prime } ( x )$
Hence show that the Maclaurin series for $\mathrm { f } ( x )$, up to and including the term in $x ^ { 2 }$, is $$\ln p + q x + r x ^ { 2 }$$ where $p , q$ and $r$ are constants to be determined.

SPS SPS FM Pure 2020 February Q2

Given that $$f ( x ) = \tan ^ { - 1 } ( x + 1 )$$ find $f ( 0 )$ and $f ^ { \prime } ( 0 )$, and show that $f ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 2 }$.
Hence find the first three terms in the Maclaurin series for $f ( x )$