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

1. 1
   The curve $C_1$ has equation $y=x^{3}e^{2x}$.

   (a)

   Find $\dfrac{dy}{dx}$.

   [1 mark]

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

   (b)

   Find the $x$-coordinate of the stationary point of $C_1$ with $x\ne0$.

   [1 mark]

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

   (c)

   Find the exact gradient of $C_1$ at $x=1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C_2$ has equation $y=\dfrac{e^{3x}}{x}$ for $x>0$.

   (a)

   Find $\dfrac{dy}{dx}$.

   [1 mark]

   • A$\dfrac{e^{3x}(3x+1)}{x^{2}}$
   • B$\dfrac{3e^{3x}}{x^{2}}$
   • C$\dfrac{e^{3x}(3x-1)}{x^{2}}$
   • D$\dfrac{e^{3x}(1-3x)}{x^{2}}$

   (b)

   Find the $x$-coordinate of the stationary point of $C_2$.

   [1 mark]

   • A$\frac13$
   • B$3$
   • C$-\frac13$
   • D$0$

   (c)

   Find the exact $y$-coordinate of the stationary point of $C_2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two curves have equations $C_3:\ y=\cos\left(x^{2}\right)$ and $C_4:\ y=\tan^{2}2x$, where $x$ is in radians.

   (a)

   Find $\dfrac{dy}{dx}$ on $C_3$.

   [3 marks]

   (b)

   Show that the gradient of $C_4$ at $x=\dfrac{\pi}{8}$ is $8$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_5$ has equation $y=x^{2}\ln x$ for $x>0$.

   (a)

   Find the exact coordinates of the stationary point of $C_5$.

   [6 marks]

   (b)

   Find the equation of the tangent to $C_5$ at the point where $x=e$, giving your answer in the form $y=mx+c$. Find the exact $x$-coordinate of the point where this tangent meets the $x$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions