Product Rule Calculator

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

About Product Rule Calculator

The product rule states (uv)' = u'v + uv'. Both terms are needed because changing either factor changes the product, and the total rate is the sum of the two effects.

Formula

Product rule

(uv)' = u'v + uv'

Worked example

With the default values below, Product Rule Calculator returns (uv)' = u'v + uv' of 1.97260236. Change any input above and every figure here is recalculated in your browser.

Inputs used

First factorone of a·xᵖ, a·sin(kx), a·cos(kx), a·e^(kx), a·ln(kx): a·xᵖ
Second factorone of a·xᵖ, a·sin(kx), a·cos(kx), a·e^(kx), a·ln(kx): a·sin(kx)
Power p for any power factor: 2
Evaluate at x: 2

Results

(uv)' = u'v + uv': 1.97260236
u'v: 3.63718971
uv': -1.66458735
Value of the product uv: 3.63718971

Step by step

1Values used
2Product rule
3(uv)' = u'v + uv'= 1.97260236
4u'v= 3.63718971
5uv'= -1.66458735
6Value of the product uv= 3.63718971

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 x²·sin x at x = 2.

2x·sin x + x²·cos x = 4 sin 2 + 4 cos 2 ≈ 3.6372624 − 1.6645873 ≈ 1.9726751.

Is (uv)' equal to u'v'?

No, and that is the most common mistake. Only the sum u'v + uv' is correct.