Question Bank

Differentiate $y = \sqrt{3x^2 - 1}$.

*You do not need to simplify your answer.*

Find the rate of change of the function $f(t) = 5\ln(3t - 1)$ when $t = 4$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

Find the gradient of the tangent to the curve $y = \dfrac{e^{2x}}{1+x^2}$ at the point where $x = 2$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

For what value(s) of $x$ is the function $y = x^3 e^x$ decreasing?

You must use calculus and show any derivatives that you need to find when solving this
problem.

The volume of a sphere is increasing.

At the instant when the sphere’s radius is $0.5$ m, the surface area of the sphere is increasing at
a rate of $0.4$ m$^2$ s$^{-1}$.

Find the rate at which the volume of the sphere is increasing at this instant.

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Differentiate $y = (2x - 5)^4$.

*You do not need to simplify your answer.*

Find the gradient of the tangent to the curve $y = \tan 2x$ at the point on the curve where $x = \dfrac{\pi}{6}$.

You must use calculus and show any derivatives that you need to find when solving this
problem.

A curve is defined parametrically by the equations $x = \dfrac{1}{(5-t)^2}$ and $y = 5t - t^2$.

Find the gradient of the tangent to the curve at the point when $t = 2$.

*You must use calculus and show any derivatives that you need to find when solving this problem.*

If $y = e^u$ and $u = \sin 2x$ show that

$\dfrac{d^2 y}{dx^2} = \dfrac{d^2 y}{du^2}\left(\dfrac{du}{dx}\right)^2 + \dfrac{dy}{du}\cdot\dfrac{d^2 u}{dx^2}$

*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Differentiate $y = \dfrac{4}{\sin x}$.

*You do not need to simplify your answer.*