All subjects · Further Mathematics · WFM03 Further Pure Mathematics 3

Differentiating inverse functions

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

The inverse trigonometric and inverse hyperbolic functions have algebraic derivatives. You must be able to derive them by implicit differentiation and use them with the chain and product rules.

Practise this in the app: instant marking and a fix for every mistake. Free, no email needed to start.

Practise this lesson

The method

Let y = arsinh x. Then x = sinh y, so dx/dy = cosh y = sqrt(1 + sinh2 y) = sqrt(1 + x2).

So dy/dx = 1/sqrt(1 + x2).

For y = arcosh x: dx/dy = sinh y = sqrt(cosh2 y - 1) = sqrt(x2 - 1), positive because y ≥ 0.

For y = arcsin x: dx/dy = cos y = sqrt(1 - x2), positive because -π/2 ≤ y ≤ π/2.

The results

These are in the formula booklet, but you may be asked to prove them.

ydy/dx
arcsin x1/sqrt(1 - x2)
arccos x-1/sqrt(1 - x2)
arctan x1/(1 + x2)
arsinh x1/sqrt(x2 + 1)
arcosh x1/sqrt(x2 - 1), x > 1
artanh x1/(1 - x2), |x| < 1

Using them

The chain rule factor is the derivative of the inside function.

Worked example

Prove that d/dx(arctan x) = 1/(1 + x2).

  1. Let y = arctan x, so x = tan y
  2. dx/dy = sec2 y = 1 + tan2 y = 1 + x2
  3. dy/dx = 1/(dx/dy)

Answer: dy/dx = 1/(1 + x2)

Key idea: Write x = f(y), find dx/dy, express it in x with an identity, then flip. arcsin: 1/sqrt(1 - x2); arctan: 1/(1 + x2); arsinh: 1/sqrt(x2 + 1); arcosh: 1/sqrt(x2 - 1); artanh: 1/(1 - x2).

Check yourself

  1. y = arcsin(2x). Find dy/dx when x = 0.3.
    Show answer

    2.5. dy/dx = 2/sqrt(1 - 4x2) = 2/sqrt(1 - 0.36) = 2/0.8 = 2.5.

  2. y = arctan(x2). Find dy/dx when x = 1.
    Show answer

    1. dy/dx = 2x/(1 + x4) = 2/2 = 1.

  3. y = (1/2) artanh(x2). Find the exact value of dy/dx when x = 1/2. Give your answer as a fraction in its simplest form.
    Show answer

    8/15. dy/dx = x/(1 - x4) = (1/2)/(1 - 1/16) = (1/2)/(15/16) = 8/15.

  4. y = arsinh(3x). Find dy/dx when x = 2/3, to 3 decimal places.
    Show answer

    1.342. dy/dx = 3/sqrt(9x2 + 1) = 3/sqrt(5) = 1.342.

  5. What is d/dx(arcosh x), for x > 1?
    • 1/sqrt(x2 - 1)
    • 1/sqrt(1 - x2)
    • -1/sqrt(x2 - 1)
    • 1/(x2 - 1)
    Show answer

    1/sqrt(x2 - 1). x = cosh y, dx/dy = sinh y = sqrt(x2 - 1), positive since y ≥ 0.

  6. Why is d/dx(arccos x) negative?
    • arccos x is always negative
    • cos x is an odd function
    • arccos x is a decreasing function
    • The square root is negative
    Show answer

    arccos x is a decreasing function. As x increases from -1 to 1, arccos x falls from π to 0, so its gradient is negative.

  7. y = x arctan x. Find dy/dx when x = 1, to 3 decimal places.
    Show answer

    1.285. dy/dx = arctan x + x/(1 + x2). At x = 1: π/4 + 1/2 = 1.285.

Practise with instant feedback

Sources

Last updated 8 November 2026. Spotted a mistake? Tell us.