All subjects · Mathematics · WMA13 (P3)

The quotient rule and sec, cosec, cot

The quotient rule differentiates one function divided by another. It also gives the derivatives of tan, sec, cosec and cot.

The rule

If y = u/v, then dy/dx = (v du/dx - u dv/dx)/v2.

The order on top matters: v u' comes first. Write u, v, u', v' first, as for the product rule.

Trig results

Learn these and be able to prove them with the quotient rule.

Tidy answers

Simplify the top and factorise. For a stationary point, set the numerator to zero: the denominator v2 cannot make a fraction zero.

Worked example

Differentiate y = ex/x.

  1. u = ex, u' = ex; v = x, v' = 1
  2. dy/dx = (xex - ex)/x2

Answer: dy/dx = ex(x - 1)/x2

Key idea: y = u/v gives (v u' - u v')/v2. sec x gives sec x tan x; cosec x gives -cosec x cot x; cot x gives -cosec2 x.

Check yourself

  1. Differentiate x/(x + 1).
    • -1/(x + 1)2
    • (2x + 1)/(x + 1)2
    • 1/(x + 1)2
    • 1
    Show answer

    1/(x + 1)2. ((x + 1)(1) - x(1))/(x + 1)2 = 1/(x + 1)2.

  2. Find the gradient of y = (2x + 3)/(x - 1) at x = 2.
    Show answer

    -5. dy/dx = (2(x - 1) - (2x + 3))/(x - 1)2 = -5/(x - 1)2. At x = 2: -5.

  3. Differentiate sec x.
    • tan2 x
    • -sec x tan x
    • sec x tan x
    • cosec x
    Show answer

    sec x tan x. sec x = (cos x)-1, so d/dx = -(cos x)-2(-sin x) = sin x/cos2 x = sec x tan x.

  4. Which expression is d/dx (cot x)?
    • cosec2 x
    • -cosec2 x
    • -sec2 x
    • -cosec x cot x
    Show answer

    -cosec2 x. cot x = cos x/sin x. The quotient rule gives (-sin2 x - cos2 x)/sin2 x = -1/sin2 x.

  5. Find the gradient of y = sin x/x at x = π, to 3 decimal places.
    Show answer

    -0.318. dy/dx = (x cos x - sin x)/x2. At x = π: (π(-1) - 0)/π2 = -1/π = -0.318.

  6. Find the x-coordinate of the stationary point of y = ln x/x, to 3 decimal places.
    Show answer

    2.718. dy/dx = (1 - ln x)/x2 = 0 gives ln x = 1, so x = e = 2.718.

  7. Find the gradient of y = cosec 2x at x = π/12, to 3 significant figures.
    Show answer

    -6.93. dy/dx = -2cosec 2x cot 2x. At x = π/12: 2x = π/6, cosec = 2 and cot = sqrt(3), so dy/dx = -4sqrt(3) = -6.93.

  8. How can you show that d/dx (tan x) = sec2 x?
    • Write tan x as sin x × cos x and use the product rule
    • Differentiate sin x and cos x and divide the results
    • Write tan x as sin x/cos x and use the quotient rule
    • Use the chain rule on tan2 x
    Show answer

    Write tan x as sin x/cos x and use the quotient rule. (cos x cos x - sin x(-sin x))/cos2 x = 1/cos2 x = sec2 x.

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Last updated 7 November 2026. Spotted a mistake? Tell us.