Question Bank

Differentiate $y = (\ln x)^2$.

*You do not need to simplify your answer.*

Find the $x$-value(s) of any stationary points on the graph of the function $f(x)=\dfrac{x^2+1}{x}$.

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:

$x = 2 + 3t$ and $y = 3t - \ln(3t - 1)$ where $t > \dfrac{1}{3}$.

Find the coordinates, $(x,y)$, of any point(s) on the curve where the tangent to the curve has a gradient
of $\dfrac{1}{2}$.

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

If $p$ is a positive real constant, prove that $y = e^{px^2}$ does not have any points of inflection.

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

Differentiate $f(x) = (5x - 3)\sin(4x)$.

*You do not need to simplify your answer.*

Find the gradient of the tangent to the curve $y = (3x^2 - 2)^3$ when $x = 2$.

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

An object is travelling in a straight line. Its displacement, in metres, is given by the formula:

$d(t) = \dfrac{t^2 - 6}{2t^3}$ where $t > 0$, $t$ is time in seconds.

Find the time(s) when the object is stationary.

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

Differentiate $y = e^{4\sqrt{x}}$.

You do not need to simplify your answer.