Differentiating inverse trigonometric and hyperbolic functionsEdexcel International A Level Further Maths: Subtopic test

1. 1
   Let $y=\operatorname{arsinh}x$, so that $\sinh y=x$.

   (a)

   Find $\frac{dx}{dy}$ in terms of $y$.

   [1 mark]

   • A$\sinh y$
   • B$-\cosh y$
   • C$\frac{1}{\cosh y}$
   • D$\cosh y$

   (b)

   Find the gradient of the curve $y=\operatorname{arsinh}x$ at $x=\frac34$.

   [1 mark]

   • A$\frac45$
   • B$\frac54$
   • C$\frac{4}{\sqrt7}$
   • D$\frac35$

   (c)

   Show that $\frac{dy}{dx}=\frac{1}{\sqrt{1+x^2}}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=x\arctan x$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$\arctan x+\frac{1}{1+x^2}$
   • B$\frac{x}{1+x^2}$
   • C$\arctan x+\frac{x}{1+x^2}$
   • D$\arctan x+\frac{x}{\sqrt{1-x^2}}$

   (b)

   Find the exact value of $f'(1)$.

   [1 mark]

   • A$\frac\pi4+1$
   • B$\frac\pi4+\frac12$
   • C$\frac12$
   • D$\frac\pi2+\frac12$

   (c)

   Show that $f$ has exactly one stationary point.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=\arcsin x+x\sqrt{1-x^2}$ for $-1<x<1$.

   (a)

   Show that $\frac{dy}{dx}=2\sqrt{1-x^2}$.

   [3 marks]

   (b)

   Find an equation of the tangent to $C$ at the point where $x=\frac12$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x)=x\operatorname{arsinh}x-\sqrt{1+x^2}$.

   (a)

   Show that $f'(x)=\operatorname{arsinh}x$.

   [6 marks]

   (b)

   Find an equation of the normal to the curve $y=f(x)$ at the point where $x=\frac34$. Give your answer in terms of $\ln2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions