7. Find the gradient of $$y=\left(x-\dfrac{\pi}{2}\right)^5\sin2x$$ in terms of $\pi$ when $x = \dfrac{3\pi}{2}$.
$-2\pi^5$
8. Find the gradient of $$y=(2\sin x)\ln(3x)$$ in terms of $\pi$ when $x = \pi$.
$-2\ln(3\pi)$
9. Find the exact gradient of $$y = x^4\mathrm{e}^{7x-3}$$ when $x = 1$.
$11\mathrm{e}^4$
10. Find the exact gradient of $$y = (x+3)^2e^{3x}$$ when $x=2$.
$85\mathrm{e}^6$
11. Find the equation of the tangent to $$y=\dfrac{1}{x}\mathrm{e}^{\frac{1}{3}x}$$ when $x=3$.
$y = \dfrac{1}{3}\mathrm{e}$
12. Find the equation of the normal to $$y=\dfrac{\mathrm{e}^x}{3+2x}$$ at the point where the curve crosses the $y$ axis.
$y = -9x+\dfrac{1}{3}$
13. Find the equation of the tangent to $$y=\dfrac{\mathrm{e}^{2x}}{(x-2)^2}$$ when $x=0$.
$y = \dfrac{3}{4}x+\dfrac{1}{4}$
14. Find the equation of the normal to $$y=2\ln(2x+5)-2x$$ when $x=-1.5$.
$x=-1.5$
15. Find the exact value of the gradient of $$y=\dfrac{\ln x}{\sin 3x}$$ when $x=\dfrac{\pi}{9}$.
$\dfrac{6\sqrt{3}-2\pi\ln(\frac{\pi}{9})}{\pi}$
16. 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.
$9.62$
17. Differentiate $$y=\ln\left(\dfrac{\mathrm{e}^x+1}{\mathrm{e}^x-1}\right)$$ with respect to $x$.
$\dfrac{2\mathrm{e}^x}{1-\mathrm{e}^{2x}}$
18. 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 $c$.
