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Chain Rule Calculator

Work out chain rule instantly with clear inputs, formula shown and shareable results.

About Chain Rule Calculator

The chain rule differentiates a composition by multiplying the outer derivative, evaluated at the inner function, by the inner derivative: d/dx f(u(x)) = f'(u)·u'(x).

Formula

Chain rule

d/dx f(u(x)) = f'(u(x)) · u'(x)

Worked example

With the default values below, Chain Rule Calculator returns dy/dx = f'(u)·u'(x) of -5.93995498. Change any input above and every figure here is recalculated in your browser.

Inputs used

Outer functionone of sin(u), cos(u), e^u, ln(u), √u
sin(u)
Inner function a·x^p: coefficient a
3
Inner power p
2
Evaluate at x
1

Results

dy/dx = f'(u)·u'(x)
-5.93995498
Inner value u
3.00000000
f'(u)
-0.98999250
u'(x)
6.00000000

Step by step

  1. 1Values used
  2. 2Chain rule
  3. 3dy/dx = f'(u)·u'(x)= -5.93995498
  4. 4Inner value u= 3.00000000
  5. 5f'(u)= -0.98999250
  6. 6u'(x)= 6.00000000

Assumptions and limits

  • The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Differentiate sin(3x²) at x = 1.

u = 3, u' = 6, so the derivative is 6·cos 3 ≈ −5.9399666.

How do I handle three nested functions?

Apply the rule repeatedly: the derivatives of all the layers multiply together.

Related Calculators

Chain Rule Calculator – Calculate dy/dx = f'(u)·u'(x)