Differentiating standard functionsEdexcel A-Level Maths: Subtopic test

1. 1
   A curve has equation $y=e^{3x}+\sin 2x$, where $x$ is in radians.

   (a)

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

   [1 mark]

   • A$3e^{3x}+\cos 2x$
   • B$3e^{3x}+2\cos 2x$
   • C$3e^{3x}-2\cos 2x$
   • D$\frac13e^{3x}-\frac12\cos 2x$

   (b)

   Find the gradient of the curve at the point where $x=0$.

   [1 mark]

   • A$3$
   • B$4$
   • C$1$
   • D$5$

   (c)

   Find the equation of the tangent to the curve at the point where $x=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f$ is defined by $f(x)=5^{2x}$ for all real $x$.

   (a)

   Find $f'(x)$.

   [1 mark]

   • A$2\ln5\times5^{2x}$
   • B$5^{2x}\ln5$
   • C$2x\times5^{2x-1}$
   • D$2\times5^{2x}$

   (b)

   Find the exact gradient of the graph of $y=f(x)$ at the point where $x=0$.

   [1 mark]

   • A$2$
   • B$\ln5$
   • C$2\ln5$
   • D$10$

   (c)

   Find the exact value of $x$ for which $f'(x)=50\ln5$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $y=\sin x$ (with $x$ in radians) has a point $P$ with $x$-coordinate $x$. A point $Q$ on the curve has $x$-coordinate $x+h$, where $h$ is small and non-zero.

   (a)

   Show that the gradient of the chord $PQ$ is $\sin x\left(\frac{\cos h-1}{h}\right)+\cos x\left(\frac{\sin h}{h}\right)$.

   [3 marks]

   (b)

   Use the small-angle approximations $\sin h\approx h$ and $\cos h\approx1-\frac{h^2}{2}$ to deduce the gradient of $y=\sin x$ at $P$ as $h$ tends to $0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ has equation $y=3\ln x-\cos 2x$ for $x>0$, where $x$ is in radians.

   (a)

   (i) Find $\frac{dy}{dx}$.
   (ii) Show that $C$ has no stationary point for $0<x\le\frac{\pi}{2}$.

   [6 marks]

   (b)

   Find the equation of the tangent to $C$ at the point where $x=\frac{\pi}{4}$, giving your answer in the form $y=mx+c$ with $m$ and $c$ in exact form.

   [6 marks]

   Total for question 4: 12 marks

End of questions