Differentiation – Year 12 A level maths worksheet with answers to download as PDF

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

1. \(y=5x^3-2x^2+7x-4\) [2 marks]
2. \(y=3\sqrt x-\dfrac2{x^2}\) [3 marks]
3. \(y=(x+2)(x-3)\) [3 marks]
4. \(y=\dfrac{2x^2-3}{x}\) [3 marks]

Differentiate from first principles.

1. \(f(x)=3x^2\) [4 marks]
2. \(f(x)=x^3\) [4 marks]

Find the equations of the tangent and the normal to each curve at the given point.

1. \(y=x^2-3x+1\) at \(x=2\) [5 marks]
2. \(y=2x^3-x\) at \(x=1\) [5 marks]

\(f(x)=x^3-6x^2+9x+2\).

1. Find \(f'(x)\) and factorise it. [3 marks]
2. Find the values of \(x\) for which \(f\) is increasing. [3 marks]
3. Find the values of \(x\) for which \(f\) is decreasing. [1 mark]

\(y=x^3-3x^2-9x+5\).

1. Find \(\dfrac{dy}{dx}\) and the \(x\)-coordinates of the stationary points. [3 marks]
2. Find the \(y\)-coordinates. [2 marks]
3. Use \(\dfrac{d^2y}{dx^2}\) to determine the nature of each stationary point. [3 marks]

\(f(x)=x^4-8x^2\).

1. Find \(f'(x)\) and \(f''(x)\). [3 marks]
2. Find the stationary points and their nature. [5 marks]
3. Find the range of values of \(x\) for which \(f\) is increasing. [3 marks]

A ball is thrown upwards from a height of 1.2 m. Its height \(h\) metres after \(t\) seconds is \(h=1.2+14t-4.9t^2\).

1. Find \(\dfrac{dh}{dt}\) and explain what it represents. [3 marks]
2. Find the time and the height at the highest point. [4 marks]
3. Find the time at which the ball lands, to 2 decimal places. [3 marks]
4. Find the velocity on landing. [2 marks]

A company makes an open-topped box with a square base of side \(x\) metres and height \(h\) metres. The volume must be 32 m\(^3\).

1. Show that the surface area is \(S=x^2+\dfrac{128}x\). [3 marks]
2. Find the value of \(x\) that minimises \(S\). [4 marks]
3. Show that this is a minimum and find the minimum surface area. [3 marks]
4. The material costs £15 per square metre. Find the minimum cost. [1 mark]

A small firm makes \(x\) hundred items a week. Its weekly profit, in hundreds of pounds, is modelled by \(P=-2x^2+60x-300\).

1. Find \(\dfrac{dP}{dx}\). [1 mark]
2. Find the production level giving the greatest profit and the profit at that level in pounds. [4 marks]
3. Explain why \(x=15\) is a maximum. [2 marks]
4. For which values of \(x\) is the firm making a profit? Give answers to 2 decimal places. [4 marks]

A particle moves on a straight line so that its displacement from the origin after \(t\) seconds is \(s=t^3-9t^2+24t\) metres.

1. Find the velocity \(v=\dfrac{ds}{dt}\) and the acceleration \(a=\dfrac{dv}{dt}\). [3 marks]
2. Find the times when the particle is instantaneously at rest. [3 marks]
3. Find the displacement at each of these times. [2 marks]
4. Find the acceleration at each of these times. [2 marks]

Part of a rollercoaster track is modelled by \(y=x^3-6x^2+9x\), \(0\le x\le4\), with \(x\) and \(y\) in tens of metres.

1. Find \(\dfrac{dy}{dx}\) and the gradient at \(x=0.5\). [3 marks]
2. Find the stationary points. [4 marks]
3. Show that the gradient is a minimum when \(x=2\) and find its value. [4 marks]
4. Explain what this means for the ride. [1 mark]

A cylindrical drinks can holds 330 ml (330 cm\(^3\)). The radius is \(r\) cm and the height is \(h\) cm.

1. Show that the total surface area is \(S=2\pi r^2+\dfrac{660}r\). [3 marks]
2. Find the radius that minimises \(S\), to 2 decimal places. [4 marks]
3. Find the height and the minimum surface area. [3 marks]
4. What do you notice about \(h\) and \(r\)? [1 mark]

A footpath follows the curve \(y=x^2-4x+5\) (units: 10 m). A lamp is to be placed on the normal to the path at \(P(3,2)\).

1. Show that \(P\) lies on the curve and find the gradient there. [2 marks]
2. Find the equation of the normal at \(P\). [3 marks]
3. The normal meets the curve again at \(Q\). Find the coordinates of \(Q\). [4 marks]
4. Find the distance \(PQ\). [2 marks]

The population of a town (in hundreds) is modelled by \(P=2000+150t-3t^2\) where \(t\) is the number of years since 2000.

1. Find the rate of change of the population at \(t=10\). [2 marks]
2. When does the population stop growing? [2 marks]
3. What is the maximum population? [2 marks]
4. Find when the model predicts the population returns to its 2000 value. [2 marks]

A rectangular sheet of card measures 24 cm by 15 cm. A square of side \(x\) cm is cut from each corner and the sides are folded up to make an open tray of volume \(V\) cm\(^3\).

1. Show that \(V=4x^3-78x^2+360x\). [3 marks]
2. Check the table for \(x=3\). [1 mark]
3. Find \(\dfrac{dV}{dx}\) and the value of \(x\) that gives the maximum volume. [5 marks]
4. Use the second derivative to prove it is a maximum and state the maximum volume. [3 marks]

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

1. Find \(\dfrac{dy}{dx}\) and the equation of the tangent at the point \(A\) where \(x=2\). [5 marks]
2. Show that the tangent meets \(C\) again where \(x^3-12x+16=0\). [2 marks]
3. Factorise \(x^3-12x+16\) and find the other point \(B\) where the tangent meets \(C\). [5 marks]

The curve \(y=x^3+ax^2+bx+c\) has stationary points at \(x=-2\) and \(x=1\) and passes through the point \((1,-2.5)\).

1. Show that \(\dfrac{dy}{dx}=3(x+2)(x-1)\) and hence find \(a\) and \(b\). [4 marks]
2. Find \(c\). [2 marks]
3. Find the coordinates of the stationary points and their nature. [4 marks]
4. State the range of values of \(x\) for which \(y\) is decreasing. [1 mark]

A rectangular allotment must have an area of 100 m\(^2\). The width is \(x\) m.

1. Show that the perimeter is \(P=2x+\dfrac{200}x\). [2 marks]
2. Check two entries in the table. [2 marks]
3. Use differentiation to find the minimum perimeter and the dimensions. [5 marks]
4. The fencing costs £8.40 per metre. Find the minimum cost. [1 mark]

\(f(x)=x^4-4x^3\).

1. Show that \(f'(x)=4x^2(x-3)\). [2 marks]
2. Find the stationary points. [3 marks]
3. Determine the nature of each using the sign of \(f'(x)\) either side. [4 marks]
4. Find \(f''(x)\) and the values of \(x\) where \(f''(x)=0\). [3 marks]

The point \(P(x,x^2)\) lies on the parabola \(y=x^2\). The point \(A\) is \((0,3)\).

1. Show that \(AP^2=x^4-5x^2+9\). [3 marks]
2. Find the values of \(x\) for which \(AP^2\) is stationary and determine their nature. [6 marks]
3. Find the least distance from \(A\) to the parabola in exact form. [3 marks]