Product Rule Calculator

The product rule states $\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$. You cannot simply multiply the derivatives. A useful mnemonic: “derivative of the first times the second, plus the first times the derivative of the second.”

For three factors: $(fgh)' = f'gh + fg'h + fgh'$. In general, for $n$ factors there are $n$ terms, each differentiating exactly one factor and leaving the rest unchanged.

The quotient rule is $\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$. It is derived from applying the product rule to $f \cdot g^{-1}$. The mnemonic “lo d-high minus high d-low, over the square of what's below” helps remember the formula.

Use logarithmic differentiation when you have products of many factors or expressions like $x^x$. Taking the log converts products to sums: $\ln(y) = \ln(f) + \ln(g) + \ldots$, making differentiation simpler.

Sometimes. For products of polynomials like $(x+1)(x-1) = x^2 - 1$, expanding first and using the power rule is simpler. But for products involving trig, exponential, or logarithmic functions (like $x^2 \cdot \sin(x)$), the product rule is unavoidable.