Product, quotient and chain rulesEdexcel A-Level Maths: Subtopic test

1. 1
   The function $f$ is defined by $f(x)=x^{3}e^{2x}$ for all real $x$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$6x^{2}e^{2x}$
   • B$3x^{2}e^{2x}+x^{3}e^{2x}$
   • C$x^{2}e^{2x}(2x+3)$
   • D$x^{2}e^{2x}(3-2x)$

   (b)

   Find the non-zero $x$-coordinate of the stationary point of the curve $y=f(x)$.

   [1 mark]

   • A$-\frac32$
   • B$\frac32$
   • C$-\frac23$
   • D$-3$

   (c)

   Find the equation of the tangent to the curve $y=f(x)$ at the point where $x=1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $h$ is defined by $h(x)=\frac{\ln x}{x^{2}}$ for $x>0$.

   (a)

   Find $h'(x)$.

   [1 mark]

   • A$\frac{2\ln x-1}{x^{3}}$
   • B$\frac{1}{2x^{2}}$
   • C$\frac{1+2\ln x}{x^{3}}$
   • D$\frac{1-2\ln x}{x^{3}}$

   (b)

   Find the $x$-coordinate of the stationary point of the curve $y=h(x)$.

   [1 mark]

   • A$e^{2}$
   • B$\sqrt e$
   • C$\frac12$
   • D$\frac e2$

   (c)

   Find the exact value of $h(x)$ at the stationary point.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A spherical balloon is inflated so that its volume, $V$ cm$^3$, increases at a constant rate of $200$ cm$^3$ s$^{-1}$. The radius of the balloon is $r$ cm. The volume of a sphere is $V=\frac43\pi r^{3}$ and its surface area is $S=4\pi r^{2}$.

   (a)

   Find the exact rate of increase of the radius when $r=5$.

   [3 marks]

   (b)

   Find the rate at which the surface area of the balloon is increasing when $r=5$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ is given by $x=3\tan2y$ for $-\frac{\pi}{4}<y<\frac{\pi}{4}$. A second curve $D$ has equation $y=\frac{\cos x}{1+\sin x}$ for $0\le x\le\frac{\pi}{2}$.

   (a)

   (i) Find $\frac{dx}{dy}$.
   (ii) Hence show that $\frac{dy}{dx}=\frac{3}{2(9+x^{2})}$ for curve $C$.
   (iii) Find the gradient of $C$ at the origin.

   [6 marks]

   (b)

   (i) Show that $\frac{dy}{dx}=-\frac{1}{1+\sin x}$ for curve $D$.
   (ii) Find the gradient of $D$ at $x=\frac\pi6$.

   [6 marks]

   Total for question 4: 12 marks

End of questions