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3.2

The Product and Quotient Rules

The Product Rule

$$\begin{gathered} \frac{d}{dx}\left[f\right(x\left)g\right(x\left)\right] = \\ f\left(x\right)\frac{d}{dx}\left[g\right(x\mathrm{)}] + g(x\left)\frac{d}{dx}\right[f\left(x\right)] \end{gathered}$$

If f and g are both differentiable. The Product Rule says that the derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

The Quotient Rule

$$\begin{gathered} \frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right] = \\ \frac{g\left(x\right){\displaystyle \frac{d}{dx}}\left[f\right(x\left)\right]-\left[f\right(x\left)\right]{\displaystyle \frac{d}{dx}}\left[g\right(x\left)\right]}{\left[g\right(x){]}^2} \end{gathered}$$

If f and g are both differentiable. The Quotient rule says that the derivative of a quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Links

• Home
• 3.1: Derivatives of Polynomials and Exponential Functions
• 3.2: The Product and Quotient Rules
• 3.3: Derivatives of Trigonometric Functions
• 3.4: The Chain Rule
• 3.5: Implicit Differentiation
• 3.6: Steps in Logarithmic Differentiation
• 3.7: Rates of Change in the Natural and Social Sciences
• 3.8: Exponential Growth and Decay
• 3.9: Related Rates
• 3.10: Linear Approximations and Differentials
• 3.11: Hyperbolic Functions