1. Why These Rules Are Needed
When two functions are multiplied or divided, their derivatives cannot be found by simply differentiating each part independently. The product and quotient rules capture how both functions change together and give the correct derivative.
2. Product Rule
The product rule handles derivatives of expressions where two functions are multiplied. It considers how both functions vary at the same time.
2.1. Formula
\frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + f'(x)g(x)
2.2. How to Remember
Think: first × derivative of second + second × derivative of first.
2.3. Example 1 — Polynomial × Polynomial
f(x) = x^2, \quad g(x) = x^3
\frac{d}{dx}[x^2 \cdot x^3] = x^2(3x^2) + (2x)(x^3) = 5x^4
2.4. Example 2 — Polynomial × Exponential
f(x) = x^2 e^x
f'(x) = x^2 e^x + 2x e^x = e^x(x^2 + 2x)
3. Quotient Rule
The quotient rule is used when one function is divided by another. It works by tracking how both numerator and denominator change.
3.1. Formula
\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}
3.2. How to Remember
Bottom × derivative of top − top × derivative of bottom, all divided by bottom squared.
3.3. Example 1 — Simple Rational Function
f(x) = \frac{x^2}{x+1}
f'(x) = \frac{(x+1)(2x) - x^2(1)}{(x+1)^2}
3.4. Example 2 — Trig over Polynomial
f(x) = \frac{\sin x}{x}
f'(x) = \frac{x \cos x - \sin x}{x^2}
4. When to Simplify Before Differentiating
Sometimes an expression is easier to simplify algebraically before applying the rules. For example, if the quotient can be reduced to a product or a simpler power, the derivative becomes much cleaner.
However, if simplification is messy or impossible, the product or quotient rule directly gives the correct result.