Differentiation — practice questions

Evaluate $\displaystyle\lim_{x \to 5} \frac{x^2 - 25}{x - 5}$.

The function $g$ is defined by $g(x) = \dfrac{2x^2 - x - 6}{x - 2}$. State the domain of $g$, evaluate $\displaystyle\lim_{x\to 2} g(x)$, and describe the discontinuity at $x = 2$.

A function is defined by $f(x) = \begin{cases} ax + b & x < 1 \\ 3 & x = 1 \\ x^2 + ax & x > 1 \end{cases}$. Determine whether values of $a$ and $b$ exist that make $f$ continuous at $x = 1$. If so, find them; if not, prove no such values exist.

For $f(x) = x^3 - 6x^2 + 5$, find the intervals on which $f$ is decreasing.

For $f(x) = x^3 - 3x^2 - 9x$, find the intervals where $f$ is concave up and where it is concave down, and state the $x$-coordinate at which the concavity changes.

The function $f(x) = \begin{cases} ax^2 + bx & x \leq 2 \\ 3x - 2 & x > 2 \end{cases}$ is differentiable at $x = 2$. Find $a$ and $b$, and explain why differentiability imposes two conditions rather than one.

Differentiate $y = 4x^3 - \dfrac{2}{x} + 5\sqrt{x}$.

Find $f''(x)$ for $f(x) = \dfrac{x^4 + 3x^2}{2x}$, giving your answer in a form with no negative or fractional powers.

The curve $y = ax^2 + \dfrac{b}{x}$ has a horizontal tangent at the point $(2, 6)$. Find the exact values of $a$ and $b$, and determine whether the point is a maximum or a minimum.

Differentiate $y = e^{3x} + \ln(5x)$.

Find the exact coordinates of the stationary point of $y = xe^{-2x}$.

The curve $y = \ln x - kx$, where $k > 0$, has a maximum value of zero. Find the exact value of $k$.

Differentiate $y = 3\sin 4x - \cos(2x + 1)$.

Find all values of $x$ in $0 \leq x \leq 2\pi$ at which the curve $y = 2\sin x + \cos 2x$ has a stationary point.

Show that the curve $y = \tan x - x$ has no maximum or minimum points on the interval $-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$, and describe the behaviour of the curve at $x = 0$.

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 $t = 1$.

Find the equation of the tangent to $y = x^2 - 4x + 1$ at the point where $x = 3$.

Find the equation of the normal to $y = x\ln x$ at the point where $x = e$, giving your answer in the form $y = mx + c$.

Two tangents to the curve $y = x^2$ are drawn, one at $x = a$ and one at $x = b$, where $a \neq b$. Show that they intersect at the point $\left(\dfrac{a+b}{2},\; ab\right)$, and hence find the condition on $a$ and $b$ for the two tangents to be perpendicular.

Find the coordinates of the stationary points of $y = x^3 - 12x + 4$.

Find the stationary points of $y = x^4 - 4x^3$ and determine the nature of each, justifying your conclusions.

The curve $y = x^3 + ax^2 + bx$ has a stationary point at $(2, -8)$. Find $a$ and $b$, determine the nature of that stationary point, and find the coordinates of the other stationary point.

Find the coordinates of the point of inflection of $y = x^3 - 6x^2 + 2x - 1$.

Show that $y = x^4 + 2x^2$ has no points of inflection, despite having a stationary point where $f''$ is not obviously non-zero.

Prove that for any cubic $y = ax^3 + bx^2 + cx + d$ with $a \neq 0$, the point of inflection lies exactly midway between the two turning points whenever they exist.

A rectangular paddock is to be fenced using 240 m of fencing. Find the dimensions that maximise the enclosed area. You may assume your solution gives a maximum.

An open-topped box is made from a 30 cm by 30 cm square of card by cutting a square of side $x$ cm from each corner and folding up the sides. Find the value of $x$ that maximises the volume, and state the maximum volume.

A rectangle is inscribed in a semicircle of radius $R$, with its base on the diameter. Prove that the rectangle of maximum area has area $R^2$, and find its dimensions in terms of $R$.

The radius of a circular oil slick increases at 3 m per hour. Find the rate at which its area is increasing when the radius is 20 m.

A spherical balloon is being deflated so that its volume decreases at 40 cm³ per second. Find the rate at which its surface area is decreasing when the radius is 5 cm.

A ladder 5 m long leans against a vertical wall. The foot of the ladder slides away from the wall at a constant 0.4 m/s. Find the rate at which the top of the ladder is sliding down when the foot is 3 m from the wall, and explain what happens to this rate as the ladder approaches the ground.