Differentiation: chain, product and quotient rules – Year 13 A level maths worksheet with answers to download as PDF

Differentiate each function with respect to \(x\).

1. \(y=(2x+1)^5\) [2 marks]
2. \(y=e^{-4x}\) [1 mark]
3. \(y=\ln(5x^2+1)\) [2 marks]
4. \(y=\sin(3x^2)\) [2 marks]

Use the product rule to differentiate each function.

1. \(y=x^3e^{2x}\) [3 marks]
2. \(y=x^2\ln x\) [3 marks]
3. \(y=x\sin3x\) [3 marks]

Use the quotient rule to differentiate each function and simplify.

1. \(y=\dfrac{2x+1}{x^2+3}\) [3 marks]
2. \(y=\dfrac{e^x}{x}\) [3 marks]
3. \(y=\dfrac{\sin x}{x^2}\) [3 marks]

A curve has equation \(x^2+xy+y^2=7\).

1. Show that \(\dfrac{dy}{dx}=-\dfrac{2x+y}{x+2y}\). [4 marks]
2. Verify that \((1,2)\) lies on the curve and find the gradient there. [2 marks]
3. Find the equation of the normal at \((1,2)\). [3 marks]

A curve is given by \(x=2t+1\), \(y=t^3-12t\).

1. Find \(\dfrac{dy}{dx}\) in terms of \(t\). [2 marks]
2. Find the coordinates of the stationary points. [4 marks]
3. Find the gradient at \(t=3\). [1 mark]

The curve \(C\) has equation \(y=\ln(3x-2)\).

1. Find \(\dfrac{dy}{dx}\). [2 marks]
2. Find the equation of the tangent to \(C\) at the point where \(x=1\). [3 marks]
3. Find the equation of the normal at the same point, in the form \(ax+by=c\). [3 marks]

A scientist grows a colony of bacteria in a laboratory. The number of bacteria \(P\) at time \(t\) hours is modelled by \(P=200e^{0.3t}\).

1. Find the initial number of bacteria. [1 mark]
2. Find \(\dfrac{dP}{dt}\) and show that \(\dfrac{dP}{dt}=0.3P\). [2 marks]
3. Find the rate of growth when \(t=4\). [2 marks]
4. Find the time at which the colony is growing at 100 bacteria per hour. [3 marks]

A cup of tea is left to cool in a cafe. Its temperature \(\theta\) degrees Celsius after \(t\) minutes is modelled by \(\theta=20+70e^{-0.05t}\).

1. Find the initial temperature and the temperature the tea approaches in the long term. [2 marks]
2. Find \(\dfrac{d\theta}{dt}\) and interpret its sign. [2 marks]
3. Show that \(\dfrac{d\theta}{dt}=-0.05(\theta-20)\). [2 marks]
4. Find the rate of cooling when \(t=10\). [2 marks]

After a patient takes a tablet, the concentration of a drug in the blood is modelled by \(C=12te^{-0.5t}\) milligrams per litre, where \(t\) is the time in hours after taking the tablet, \(t\ge0\).

1. Show that \(\dfrac{dC}{dt}=6e^{-0.5t}(2-t)\). [3 marks]
2. Find the time at which the concentration is greatest, and the maximum concentration. [4 marks]
3. The drug is only effective while \(C>5\). Use the table to estimate for how long this is. [2 marks]

An ice cream seller models the number of ice creams sold per hour, \(S\), at time \(t\) hours after opening on a sunny day by \(S=\dfrac{400t}{t^2+4}\), \(t\ge0\).

1. Show that \(\dfrac{dS}{dt}=\dfrac{400(4-t^2)}{(t^2+4)^2}\). [4 marks]
2. Find the time at which sales per hour are greatest and the maximum value. [3 marks]
3. Find the sales rate when \(t=6\) and explain why the model might not be realistic for large \(t\). [3 marks]

A spherical party balloon is inflated so that its radius increases at a constant rate of 0.5 cm per second. The volume of a sphere is \(V=\tfrac43\pi r^3\) and its surface area is \(A=4\pi r^2\).

1. Find \(\dfrac{dV}{dt}\) when \(r=6\). [3 marks]
2. Find \(\dfrac{dA}{dt}\) when \(r=6\). [3 marks]
3. Explain why \(\dfrac{dV}{dt}\) is not constant even though \(\dfrac{dr}{dt}\) is constant. [1 mark]

Water leaks from a conical funnel held with its vertex downwards. The depth of water is \(h\) cm and the radius of the water surface is \(\dfrac h2\) cm. The volume of water is \(V=\tfrac13\pi r^2h\). Water leaks out at 20 cm\(^3\) s\(^{-1}\).

1. Show that \(V=\dfrac{\pi h^3}{12}\). [2 marks]
2. Find \(\dfrac{dh}{dt}\) when \(h=10\). [4 marks]
3. Explain why the depth falls more quickly as the funnel empties. [1 mark]

The path of a roller coaster is modelled by \(x=4t\), \(y=t^3-6t^2+9t\) for \(0\le t\le4\), where \(x\) and \(y\) are measured in metres and \(t\) is a parameter.

1. Find \(\dfrac{dy}{dx}\) in terms of \(t\). [3 marks]
2. Find the coordinates of the highest and lowest points of the track for \(0\le t\le4\) at which the track is horizontal. [4 marks]
3. Find the gradient of the track at the end \(t=4\). [2 marks]

The horizontal displacement of a pendulum bob from its rest position is \(x=5\cos2t\) centimetres, where \(t\) is the time in seconds.

1. Find the velocity \(v=\dfrac{dx}{dt}\). [2 marks]
2. Find the acceleration \(a=\dfrac{d^2x}{dt^2}\) and show that \(a=-4x\). [3 marks]
3. Find the first time at which the bob is at rest momentarily. [2 marks]
4. Find the maximum speed. [1 mark]

A curve has equation \(y=x^2e^{-x}\) for \(x\ge0\). It models the height (in metres) of a firework \(x\) seconds after launch scaled for the display.

1. Show that \(\dfrac{dy}{dx}=xe^{-x}(2-x)\). [3 marks]
2. Find the coordinates of the stationary points and determine their nature. [5 marks]
3. State the range of \(y\) and explain what happens as \(x\to\infty\). [2 marks]
4. Find the second derivative and use it to confirm the maximum. [3 marks]

A curve has equation \(x^2+xy+y^2=7\).

1. Show that the tangent is horizontal where \(y=-2x\). [3 marks]
2. Find the coordinates of the points where the tangent is horizontal. [4 marks]
3. Find the coordinates of the points where the tangent is vertical. [4 marks]

A curve is given by \(x=3\cos t\), \(y=2\sin t\) for \(0\le t<2\pi\).

1. Show that \(\dfrac{dy}{dx}=-\dfrac23\cot t\). [3 marks]
2. Find the equation of the tangent at \(t=\dfrac\pi4\). [4 marks]
3. Show that the Cartesian equation of the curve is \(\dfrac{x^2}9+\dfrac{y^2}4=1\). [2 marks]
4. Find the points where the tangent is parallel to the \(y\)-axis. [2 marks]

The displacement \(y\) of a spring-like system satisfies the differential equation \(\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-6y=0\).

1. Show that \(y=Ae^{2x}+Be^{-3x}\) is a solution for any constants \(A\) and \(B\). [4 marks]
2. Given that \(y=1\) and \(\dfrac{dy}{dx}=-8\) when \(x=0\), find \(A\) and \(B\). [4 marks]
3. Find the value of \(y\) when \(x=1\). [1 mark]
4. Describe what happens to \(y\) as \(x\to\infty\). [1 mark]

A manufacturer makes an open-topped cylindrical recycling container for glass of volume \(500\) cm\(^3\) with radius \(r\) cm and height \(h\) cm. The total surface area (base and curved side) is \(A=\pi r^2+2\pi rh\).

1. Show that \(A=\pi r^2+\dfrac{1000}{r}\). [3 marks]
2. Find \(\dfrac{dA}{dr}\) and the value of \(r\) for which \(A\) is stationary. [4 marks]
3. Show that this gives a minimum and find the minimum area and the corresponding height. [4 marks]

Let \(y=x^{\sin x}\) for \(x>0\).

1. By taking natural logarithms, show that \(\dfrac{dy}{dx}=x^{\sin x}\left(\cos x\ln x+\dfrac{\sin x}{x}\right)\). [4 marks]
2. Find the exact value of \(\dfrac{dy}{dx}\) when \(x=\dfrac\pi2\). [3 marks]

Find the equation of the tangent from the origin to each curve.

1. \(y=e^x\) [5 marks]
2. \(y=\ln x\) [5 marks]