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

Differentiate with respect to \(x\).

1. \(y=\ln(4x^2+1)\) [2 marks]
2. \(y=x^3e^{-2x}\), giving your answer in the form \(x^2e^{-2x}(a+bx)\) [2 marks]

\(y=\dfrac{x^2}{2x+1}\) for \(x\ne-\tfrac12\).

1. Show that \(\dfrac{dy}{dx}=\dfrac{2x(x+1)}{(2x+1)^2}\). [2 marks]
2. Find the coordinates of the stationary points. [2 marks]

A curve has equation \(x^2+2xy+3y^2=18\).

1. Show that the point \((3,1)\) lies on the curve and find \(\dfrac{dy}{dx}\) there. [3 marks]
2. Find the equation of the tangent at \((3,1)\). [1 mark]

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

1. Find \(\dfrac{dy}{dx}\) in terms of \(t\). [2 marks]
2. Find the equation of the normal at the point where \(t=1\). [2 marks]

The volume of a spherical balloon increases at 100 cm\(^3\) s\(^{-1}\). Find the rate of increase of the radius and of the surface area when \(r=5\) cm.