A function can be written as athe product of two other functions.

The product rule states that if you let \(F(x) = f(x) \cdot g(x)\), then

\[F^{\prime}(x) = \frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot \Bigg[ \frac{d}{dx}g(x) \Bigg] + g(x) \cdot \Bigg[ \frac{d}{dx}f(x) \Bigg]\]

In simple English, the derivative of a product is the first factor times the derivative of the second factor, plus the second factor times the derivative of the first factor.

The quotient rule states that

\[\text{if } Q(x) = \frac{N(x)}{D(x)}, \text{then } Q^{\prime}(x) = \frac{D(x) \cdot N^{\prime}(x) - N(x) \cdot D^{\prime}(x)}{[D(x)]^2}\]

In simple English, the derivative of a quotient is the denominaor times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.