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.

Why it is not just u’v’

This is the mistake worth understanding rather than memorising around. Take u(x) = x and v(x) = x. The product is x², whose derivative is 2x. But u’ = 1 and v’ = 1, so u’v’ = 1, which is wrong for every x except ½. The reason is that changing x nudges both factors at once, and the total change picks up a contribution from each. Write it out with a small step h and you get (u+Δu)(v+Δv) = uv + uΔv + vΔu + ΔuΔv. Divide by h and let h shrink: the last term vanishes because it is second order, and the two surviving terms are exactly uv’ and vu'.

Radians, not degrees

If you pick sine or cosine, the angle is in radians. This is not a preference. The derivative of sin(x) is cos(x) only when x is measured in radians; in degrees it is (π/180)cos(x), and every calculus formula picks up a stray constant. The calculator reports the numeric check alongside the answer so you can see the two agree.

A worked example with the log

Take u(x) = 2x³ and v(x) = ln(x), evaluated at x = 2. Then u = 16, u’ = 6x² = 24, v = ln 2 ≈ 0.693, and v’ = 1/x = 0.5. The product rule gives 24(0.693) + 16(0.5) = 16.635 + 8 = 24.635. Notice that the second term is not small: dropping it would put you 32% off, which is the usual size of the error this rule exists to prevent.