Q1
Answer
Differentiate the following with respect to $x$:
  1. $\dfrac{1}{\sin 3x}$
  2. $\sin^3x$
  1. $-\dfrac{3\cos 3x}{\sin^2 3x}$
  2. $3\cos x\sin^2 x$
Q2
Answer
Differentiate the following with respect to $x$:
  1. $\ln (x^2+3x)$
  2. $2^{7x}$
  1. $\dfrac{2x+3}{x^2+3x}$
  2. $7\ln 2 \times 2^{7x}$
Q3
Answer
Differentiate $$\csc^3x$$ with respect to $x$.
$-3\cot x\csc^3 x$
Q4
Answer
Differentiate $$x^2\cot 2x$$ with respect to $x$.
$2x\cot 2x - 2x^2\csc^2 2x$
Q5
Answer
Differentiate $$\dfrac{\ln(x^2+1)}{x^2+1}$$ with respect to $x$.
$\dfrac{2x - 2x\ln(x^2+1)}{(x^2+1)^2}$
Q6
Answer
Differentiate $$\dfrac{10\mathrm{e}^{-2x}}{1+3\mathrm{e}^{-5x}}$$ with respect to $x$.
$\dfrac{90\mathrm{e}^{-7x} - 20\mathrm{e}^{-2x}}{(1+3\mathrm{e}^{-5x})^2}$
Q7
Answer
Find the gradient of $$y=\left(x-\dfrac{\pi}{2}\right)^5\sin2x$$ in terms of $\pi$ when $x = \dfrac{3\pi}{2}$.
$-2\pi^5$
Q8
Answer
Find the gradient of $$y=(2\sin x)\ln(3x)$$ in terms of $\pi$ when $x = \pi$.
$-2\ln(3\pi)$
Q9
Answer
Find the exact gradient of $$y = x^4\mathrm{e}^{7x-3}$$ when $x = 1$.
$11\mathrm{e}^4$
Q10
Answer
Find the exact gradient of $$y = (x+3)^2e^{3x}$$ when $x=2$.
$85\mathrm{e}^6$
Q11
Answer
Find the equation of the tangent to $$y=\dfrac{1}{x}\mathrm{e}^{\frac{1}{3}x}$$ when $x=3$.
$y = \dfrac{1}{3}\mathrm{e}$
Q12
Answer
Find the equation of the normal to $$y=\dfrac{\mathrm{e}^x}{3+2x}$$ at the point where the curve crosses the $y$ axis.
$y = -9x+\dfrac{1}{3}$
Q13
Answer
Find the equation of the tangent to $$y=\dfrac{\mathrm{e}^{2x}}{(x-2)^2}$$ when $x=0$.
$y = \dfrac{3}{4}x+\dfrac{1}{4}$
Q14
Answer
Find the equation of the normal to $$y=2\ln(2x+5)-2x$$ when $x=-1.5$.
$x=-1.5$
Q15
Answer
Find the exact value of the gradient of $$y=\dfrac{\ln x}{\sin 3x}$$ when $x=\dfrac{\pi}{9}$.
$\dfrac{6\sqrt{3}-2\pi\ln(\frac{\pi}{9})}{\pi}$
Q16
Answer
The curve $$y=e^{2x}-18x+15$$ intersects the $y$ axis at $P$ and has a turning point at $Q$. Find, to 3 significant figures, the area between the curve and the $x$ axis between these points.
$9.62$
Q17
Answer
Differentiate $$y=\ln\left(\dfrac{\mathrm{e}^x+1}{\mathrm{e}^x-1}\right)$$ with respect to $x$.
$\dfrac{2\mathrm{e}^x}{1-\mathrm{e}^{2x}}$
Q18
Answer
The tangent to the curve $$y=x^2\cos (x^2)$$ at $\left(\dfrac{\sqrt{\pi}}{2},\dfrac{\pi\sqrt{2}}{8}\right)$ is $y = mx + c$. Find exact values of $m$ and $c$.
$m = \dfrac{\sqrt{2\pi}}{8}(4 - \pi)$, $c = \dfrac{\pi\sqrt{2}}{16}(\pi-2)$