Chain, Product & Quotient Rules — practice questions

Differentiate $y = (5x - 2)^7$.

Find $\dfrac{dy}{dx}$ for $y = e^{\sin 2x}$, and evaluate it exactly at $x = \dfrac{\pi}{4}$.

A curve has equation $y = \left(x^2 - 4x + k\right)^3$, where $k$ is a constant. Show that the curve has a stationary point at $x = 2$ for every value of $k$, and find the values of $k$ for which $x = 2$ is the only stationary point.

Differentiate $y = x^4\ln x$. You do not need to simplify your answer.

Find $\dfrac{dy}{dx}$ for $y = \dfrac{\sin 3x}{2x + 1}$, simplifying where a common factor allows.

The curve $y = \dfrac{x}{x^2 + a^2}$, where $a > 0$ is a constant, has a maximum point. Find its coordinates in terms of $a$, and show that the maximum value is inversely proportional to $a$.

A curve is defined by $x = t^3$ and $y = t^2 + 4t$. Find $\dfrac{dy}{dx}$ in terms of $t$.

For the curve $x = 2t$, $y = t^2 - 3t$, find $\dfrac{d^2y}{dx^2}$, showing your method clearly.

A curve is given by $x = t^2 - 2t$ and $y = t^3 - 3t$. Find the coordinates of all points where the tangent is horizontal and all points where it is vertical, and explain what happens at $t = 1$.