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Differentiation

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 12 marks

y=(2x-5)(x^2+3)

Work out \dfrac{dy}{dx}

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Expand the brackets first, then differentiate each term.

Worked solution
  1. Expand: y=2x^3+6x-5x^2-15
  2. Write in order: y=2x^3-5x^2+6x-15
  3. Differentiate each term: 6x^2-10x+6 (the -15 differentiates to 0)
  4. Answer: \dfrac{dy}{dx}=6x^2-10x+6

Question 23 marks

A curve has equation y=3x^4-5x+\dfrac{8}{x^2}

Work out the value of \dfrac{d^2y}{dx^2} when x=2

Hint

Write \dfrac{8}{x^2} as 8x^{-2}, differentiate twice, then substitute x=2

Worked solution
  1. Write as powers of x: y=3x^4-5x+8x^{-2}
  2. Differentiate: \dfrac{dy}{dx}=12x^3-5-16x^{-3}
  3. Differentiate again: \dfrac{d^2y}{dx^2}=36x^2+48x^{-4}
  4. Substitute x=2: 36\times4+\dfrac{48}{16}=144+3
  5. Answer: 147

Question 3Challenge5 marks

A curve has equation y=\dfrac{(x-4)^2}{2x}

(a)

Work out \dfrac{dy}{dx}

3 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Work out the x-coordinates of the two points on the curve where the rate of change of y with respect to x is -\dfrac{3}{2}

2 marks

Give every value, separated by commas

Hint

Do not differentiate the top and bottom separately: expand and split the fraction into separate powers of x first.

Worked solution

Part (a)

  1. Expand the top: (x-4)^2=x^2-8x+16
  2. Divide each term by 2x: y=\dfrac{x}{2}-4+\dfrac{8}{x}
  3. Write as powers of x: y=\frac12x-4+8x^{-1}
  4. Differentiate: \frac12-8x^{-2}
  5. Answer: \dfrac{dy}{dx}=\dfrac12-\dfrac{8}{x^2}

Part (b)

  1. Rate of change of y with respect to x means \dfrac{dy}{dx}
  2. \dfrac12-\dfrac{8}{x^2}=-\dfrac32
  3. \dfrac{8}{x^2}=2, so x^2=4
  4. Answer: x=2 or x=-2

More on this topic: Differentiation worksheet with full solutions

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