Power rule (standard derivative): notes and practice questions

Consider the function $f(x) = x^{3} + 5x^{2} - x + 5$.

(a) Find $f'(x)$.

(b) Find the equation of the tangent line at $x = 1$.

(c) For the equation of the normal line at $x = 1$.

Consider the functions $f(x) = \ln(x-p)$ and $g(x) = \frac{1}{4}x^2 + q$ where $p, q \in \mathbb{R}$.

(a) Find $g'(x)$.

The graphs of $f$ and $g$ have a common tangent at the point where $x = 2$.

(b) Show that $p = 1$.

(c) Hence, find the value of $q$.

A drone takes off from a platform. Its height, $h$ metres, above the platform after $t$ seconds is given by $h(t) = 6t - t^2$, for $0 \le t \le 8$. This is shown in the following diagram.

The drone lands back on the platform when $t=p$.

Find the value of $p$.

The drone reaches its maximum height when $t=q$.

Find the value of $q$.

Find the drone's maximum height above the platform.

Find the drone's vertical distance from the platform when $t=8$.

The total vertical distance travelled by the drone in the first 8 seconds is given by $d$.

Find the value of $d$.

A second drone, Drone B, takes off from the same platform. Its velocity is given by $v_B(t) = 8 - 2t$, for $t \ge 0$.

When $t = k$, the total vertical distance travelled by Drone B is equal to $d$.

Find the value of $k$.

A hollow pipe is manufactured by removing a smaller cylinder of radius $r$ from the centre of a larger cylinder of radius $3r$. Both cylinders have the same height, $h$. This is shown in the following diagram.

All lengths are measured in centimetres.

The total surface area of the hollow pipe, in cm$^2$, is given by $S$.

(a) Show that $S = 16\pi r^2 + 8\pi rh$.

The total surface area of the hollow pipe is $128\pi \text{ cm}^2$.

(b) Show that the volume of the pipe, $V$, is given by $V = 128\pi r - 16\pi r^3$.

(c) Find an expression for $\frac{dV}{dr}$.

(d) The hollow pipe has its maximum volume when $r = k\sqrt{6}$, where $k \in \mathbb{Q}^+$. Find the value of $k$.

(e) Hence, find this maximum volume, giving your answer in the form $m\pi\sqrt{6}$, where $m \in \mathbb{Q}^+$.

Consider the functions $f(x) = k - (x-h)^2$ and $g(x) = \ln(x-1) + 2$ where $h, k \in \mathbb{R}$.

The graphs of $f$ and $g$ have a common tangent at $x=2$.

(a) Find $g'(x)$.

(b) Show that $h = \frac{5}{2}$.

(c) Hence, find the value of $k$.

A particle moves in a straight line. Its velocity, $v \,\text{ms}^{-1}$, at time $t$ seconds is given by $v(t) = -t^3 + 6t^2 - 9t$, for $0 \le t \le 5$. The particle is at the origin at $t=0$.

The graph of $v$ is shown in the following diagram.

(a) Find the displacement of the particle from the origin at $t=3$.

(b) Find an expression for the acceleration of the particle.

(c) The particle is momentarily at rest at $t=0$ and again at $t=k$. Find the greatest speed of the particle in the interval $0 \le t \le k$.

(d) Find the greatest speed of the particle for $0 \le t \le 5$.

(e) Write down an expression that represents the distance travelled by the particle while its speed is increasing. Do not evaluate the expression.

The expression $\frac{5x-2}{\sqrt[3]{x}}$ can be written in the form $5x^p - 2x^q$. Write down the value of $p$ and the value of $q$.

Hence, find the value of $\int_1^8 \frac{5x-2}{\sqrt[3]{x}} \, dx$.

A function is defined by $f(x) = \frac{1}{2}x^2 + x + 4$. The following diagram shows part of the graph of $f$.

The graph has a vertex at V and intersects the y-axis at point P.

(a) Find the coordinates of the vertex V.

(b) Write down the coordinates of the y-intercept, P.

(c) The line L is the normal to the graph of $f$ at point P. Find the equation of L, giving your answer in the form $y=mx+c$.

(d) The line L intersects the graph of $f$ at a second point, Q. Calculate the distance between P and Q.

A landscape architect is designing a section of a garden path. The shape of one edge of the path can be modelled by the function $h(x) = \frac{1}{4}x^2 + \frac{1}{2}$ for $x \ge 0$, where $x$ and $h(x)$ are measured in metres.

(a) (i) Find $h^{-1}(x)$, the inverse of $h(x)$, and state its domain.

(ii) Write down the range of $h^{-1}(x)$.

(b) The graph of $h$ intersects the graph of $h^{-1}$ at two points. Find the $x$ -coordinates of these two points.

(c) Find the area enclosed by the graph of $h$ and the graph of $h^{-1}$.

(d) Find $h'(x)$.

(e) Find the value of $x$ for which the graph of $h$ and the graph of $h^{-1}$ have the same gradient.

A function, $g$, has its derivative given by $g'(x) = -2x^2 + 8x + k$, where $k \in \mathbb{R}$. The following diagram shows part of the graph of $g'$.

The graph of $g'$ has an axis of symmetry $x = q$.

(a) Find the value of $q$.

(b) The vertex of the graph of $g'$ has a y-coordinate of 10. Find the value of $k$.

(c) Find the equation of the tangent to the graph of $g'$ at $x = 0$.

The graph of $g$ has a point of inflexion at $x = c$.

(d) (i) Find the value of $c$.

(ii) Find the values of $x$ for which the graph of $g$ is concave-up. Justify your answer.

A remote-controlled drone is launched from a platform and moves horizontally in a straight line. Its velocity, $v$ m s$^{-1}$, at time $t$ seconds after launch, is given by $v(t) = t + 1 - \sqrt{4t + 16}$, for $t \ge 0$.

(a) Find the drone's initial velocity.

(b) Find the time when the drone is at rest.

(c) Find the drone's acceleration at the instant it comes to rest.

(d) Determine the time interval(s) when the drone is:

(i) slowing down

(ii) speeding up.

Differentiate each of the following expressions with respect to $x$:

$10e^x - 4\ln x + 2$

$e^{-5x}$

$x^2 e^x$

$\ln(4x^3 - 2x)$

$e^{\sin(x)}$

$\frac{\ln x}{x^3}$

$\ln(\frac{1}{x^2+1})$

$e^{x\cos(x)}$

The trajectory of a small drone flying over a landscape can be modelled by the function $f(x) = -0.5x^2 + 4x + 1$, where $x$ is the horizontal distance in meters from the launch point and $f(x)$ is the altitude in meters. The drone passes through a checkpoint A at a horizontal distance of 2 meters.

(a) (i) Find the gradient of the tangent to the drone's trajectory at checkpoint A.

(a) (ii) Hence, write down the gradient of the normal to the drone's trajectory at checkpoint A.

(b) Write down the equation of the normal to the drone's trajectory at checkpoint A.

(c) A searchlight beam is directed along the normal line found in part (b). This beam intersects the drone's trajectory again at a second point B. Find the coordinates of B.

A function is defined by $f(x) = x^3 - 4x$.

(a) Find the equation of the tangent to the graph of $f$ at the point where $x = -1$.

(b) The tangent line found in part (a) intersects the graph of $f$ at a second point, P. Find the coordinates of P.

(c) Find the exact area of the finite region enclosed by the graph of $f$ and the tangent line.

An engineer is designing an open-top rectangular container with a square base. The container must have a volume of $4\text{ m}^3$.

Let the side length of the square base be $x$ metres and the height of the container be $h$ metres.

Show that the total surface area, $S\text{ m}^2$, of the material used for the container is given by $S = x^2 + \frac{16}{x}$.

Find an expression for $\frac{dS}{dx}$.

Hence, find the exact value of $x$ for which the surface area is a local minimum or maximum.

Find an expression for $\frac{d^2S}{dx^2}$.

Use the second derivative of $S$ to justify that $S$ is a minimum when $x = 2$.

Find the minimum surface area of the container.

(a) A deep-sea submersible's vertical displacement, in metres, from a reference depth is given by $s(t) = 5 \sin(2t) - 3 \cos(t)$ for time $t$ minutes. Positive $s(t)$ indicates the submersible is above the reference depth, and negative $s(t)$ indicates it is below.

Find the submersible's vertical velocity and acceleration at any time $t$.

(b) Find the time intervals during $0 \le t \le 2\pi$ when the submersible is moving upwards.

(c) Determine the time intervals during $0 \le t \le 2\pi$ when the submersible's vertical velocity is decreasing.

(d) Calculate the total vertical distance travelled by the submersible during the time $0 \le t \le 2\pi$. Give your answer to three significant figures.

A civil engineer is designing a parabolic arch for a pedestrian bridge. The shape of the arch can be modelled by the function $f(x) = -x^2 + 7x - 5$, where $x$ is the horizontal distance in meters from one end of the bridge and $f(x)$ is the height of the arch above the ground in meters.

A support cable needs to be attached to the arch at a point A where $x=2$.

(i) Calculate the gradient of the tangent to the arch at point A.

(ii) Hence, write down the gradient of the line perpendicular to the tangent at point A.

The support cable is designed to be perpendicular to the arch at point A. Write down the equation of the line representing this support cable.

The support cable (represented by the normal line) is extended and intersects the parabolic arch again at a second point B. Find the coordinates of point B.