Calculators / Calculus 1 / Product Rule Calculator

Product Rule Calculator

Published on: March 30, 2025

This Product Rule Calculator helps you differentiate the product of two functions and shows each step clearly. It applies the product rule to expressions of the form u(x)v(x) by finding the derivative of each part and combining them correctly. This makes it useful for checking answers, understanding how the rule works, and practising calculus step by step.

Step-by-step method

  1. Set up u(x) and v(x).
  2. Use the product rule: (u·v)' = u'·v + u·v'.
  3. Differentiate u and v separately.
  4. Substitute into the product rule and simplify.

Formula:

(u·v)' = u'·v + u·v'

Example 1: x^2+1, sin(x)

Step 1 - Set up the problem.

In this problem: Identify u(x) and v(x) from your input.

u(x)=x2 + 1,v(x)=sin(x)

Step 2 - Pick the rule.

In this problem: Use the product rule.

(u·v)' = u'·v + u·v'

Step 3 - Differentiate u and v.

In this problem: Differentiate u(x) and v(x) separately.

u'(x)=2x,v'(x)=cos(x)

Step 4 - Apply the rule.

In this problem: Substitute into u'·v + u·v'.

f'(x)=u'·v + u·v'=2xsin(x) + (x2 + 1)cos(x)

Final answer: f'(x) = 2xsin(x) + (x^2 + 1)cos(x)

Example 2: (1/2)x^3, exp(x)

Step 1 - Set up the problem.

In this problem: Identify u(x) and v(x) from your input.

u(x)=
1
2
x3,v(x)=exp(x)

Step 2 - Pick the rule.

In this problem: Use the product rule.

(u·v)' = u'·v + u·v'

Step 3 - Differentiate u and v.

In this problem: Differentiate u(x) and v(x) separately.

u'(x)=
3
2
x2,v'(x)=exp(x)

Step 4 - Apply the rule.

In this problem: Substitute into u'·v + u·v'.

f'(x)=u'·v + u·v'=
1
2
x2(x + 3)exp(x)

Final answer: f'(x) = x^2(x + 3)exp(x)/2