Product Rule Calculator

When you differentiate a product of two functions, the product rule says: d/dx[u(x)v(x)] = u’(x)v(x) + u(x)v’(x). Both terms matter. Dropping one of them is the most common mistake students make on differentiation problems.

This calculator uses u(x) = ax^m as the first function. You choose the second function v(x) from five types: a monomial bx^n, sine, cosine, the natural exponential, or the natural log.

For each v(x) type, the derivative v’(x) is:

• Monomial bx^n: nbx^(n-1)
• sin(bx): b·cos(bx)
• cos(bx): -b·sin(bx)
• e^x: e^x
• ln(x): 1/x (requires x > 0)

Once u, u’, v, v’ are evaluated at your chosen x, the result is u’(x)v(x) + u(x)v’(x).

A worked example: u(x) = 3x^2, v(x) = sin(2x). Then u’(x) = 6x, v’(x) = 2cos(2x). At x = 1: u = 3, u’ = 6, v = sin(2) ≈ 0.909, v’ = 2cos(2) ≈ -0.832. Product rule: 6(0.909) + 3(-0.832) = 5.454 - 2.496 = 2.958.

The product rule extends naturally to three or more functions through repeated application. For u·v·w, differentiate any two with the product rule, then apply it again with the third. The same “one at a time” logic holds.

A common shorthand: think of the product rule as “first times derivative of second, plus second times derivative of first.” That order helps avoid dropping a term.