A-level · Differentiation

The quotient ruleExam-style questions

Differentiating one function divided by another. Why differentiating the top and bottom separately fails; where the rule comes from; the quotient rule (u'v − uv') ÷ v² with the order that matters; simplifying the top and leaving the bottom alone; stationary points and increasing intervals from the top's sign; and when rewriting is quicker than the rule.

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27 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4Level 5
  1. 1
    The curve CC has equation

    y=xx2+1y = \dfrac{x}{x^2 + 1}
    (a) Show that dydx=1−x2(x2+1)2\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - x^2}{\left(x^2 + 1\right)^2}. (3 marks)
    (b) Find the coordinates of the two stationary points of CC, and determine the nature of each. (4 marks)
    Level 4 · 7 marksTry it
  2. 2
    A curve has equation

    y=ln⁡xx2y = \dfrac{\ln x}{x^2}, x>0\quad x > 0.
    (a) Show that dydx=1−2ln⁡xx3\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{1 - 2\ln x}{x^3}. (3 marks)
    (b) Determine the exact coordinates of the stationary point of the curve, and show that it is a maximum. (4 marks)
    Level 4 · 7 marksTry it
  3. 3
    The curve CC has equation y=ln⁡xx2y = \dfrac{\ln x}{x^{2}},   x>0\; x > 0.
    (a) Show that dydx=1−2ln⁡xx3\dfrac{dy}{dx} = \dfrac{1 - 2\ln x}{x^{3}}. (3 marks)
    (b) Hence find the exact xx-coordinate of the stationary point of CC. (1 mark)
    Level 4 · 4 marksTry it
  4. 4
    Figure 1MCOxyFigure 1 shows a sketch of part of the curve CC with equation y=f(x)y = \text{f}(x), where

    f(x)=e2x2x−1x>12\text{f}(x) = \dfrac{\text{e}^{2x}}{2x - 1} \qquad x > \dfrac{1}{2}

    The curve has one turning point, the minimum point MM.
    (a) Show that

    f′(x)=A(x−1)e2x(2x−1)2\text{f}'(x) = \dfrac{A(x - 1)\text{e}^{2x}}{(2x - 1)^2}

    where AA is an integer to be found.

    (4 marks)
    (b) Find the exact coordinates of MM.

    (2 marks)

    The function h is defined by

    h(x)=1f(x)x>12\text{h}(x) = \dfrac{1}{\text{f}(x)} \qquad x > \dfrac{1}{2}
    (c) Using your answer to part (b), find the range of h.

    (2 marks)
    (d) Find the coordinates of the maximum point on the curve with equation y=3−f(2x)y = 3 - \text{f}(2x)

    (2 marks)
    Level 4 · 10 marksTry it
  5. 5
    f(x)=2x−1x2+3f(x) = \dfrac{2x - 1}{x^2 + 3}.
    (a) Show that f′(x)=−2x2+2x+6(x2+3)2f'(x) = \dfrac{-2x^2 + 2x + 6}{(x^2 + 3)^2}. (4 marks)
    (b) Hence find the exact range of values of xx for which ff is an increasing function. (4 marks)
    Level 4 · 8 marksTry it

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