Product Rule Calculator

The product rule states that the derivative of a product of two functions f(x) and g(x) is: d/dx[f(x)·g(x)] = f′(x)·g(x) + f(x)·g′(x). In words, you differentiate the first function and multiply by the second, then add the first function multiplied by the derivative of the second.

Use the product rule whenever you need to differentiate an expression that is the product of two distinct functions of x — for example x²·sin(x) or eˣ·ln(x). If the two factors can be simplified into a single function first, that may be easier, but the product rule always works.

Split the expression at the multiplication point. For example, in (3x²)(sin x), let f(x) = 3x² and g(x) = sin x. The choice is flexible — swapping f and g gives the same final derivative, since addition is commutative.

Yes. For three functions u·v·w, the derivative is u′vw + uv′w + uvw′. This generalises further: differentiate each factor in turn while keeping the others unchanged, then sum all the resulting terms.

By the power rule, d/dx[axⁿ] = a·n·xⁿ⁻¹. For example, d/dx[3x²] = 6x. This is the rule applied internally when you select the Polynomial type in this calculator.

The product rule handles f(x)·g(x), while the quotient rule handles f(x)/g(x). The quotient rule formula is [f′g − fg′] / g², which includes a subtraction and a squared denominator. Both rules derive from the same limit definition of the derivative.

The symbolic derivative d/dx[f·g] gives a formula valid for all x. Evaluating it at a particular x (e.g. x = 2) gives the instantaneous rate of change at that point — useful for tangent line slopes, optimisation problems, and physics applications.

The most common error is treating d/dx[f·g] as simply f′·g′ — that is wrong. Another mistake is forgetting to differentiate both factors separately and add the two terms. Always verify by expanding the product first (when feasible) and comparing derivatives.