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# 7.4 The Product and Quotient Rules ## Derivatives of Products **Example 1** Differentiate $f(x) = (x^2 + 1)(x^3 + 3)$ **Solution** $f'(x) = (x^2 + 1)(3x^2) + (x^3 + 3)(2x)$ $= 3x^4 + 3x^2 + 2x^4 + 6x$ $= 5x^4 + 3x^2 + 6x$ ## Product Rule If $f$ and $g$ are both differentiable, then $\frac{d}{...

# 7.4 The Product and Quotient Rules ## Derivatives of Products **Example 1** Differentiate $f(x) = (x^2 + 1)(x^3 + 3)$ **Solution** $f'(x) = (x^2 + 1)(3x^2) + (x^3 + 3)(2x)$ $= 3x^4 + 3x^2 + 2x^4 + 6x$ $= 5x^4 + 3x^2 + 6x$ ## Product Rule If $f$ and $g$ are both differentiable, then $\frac{d}{dx}[f(x)g(x)] = f(x)\frac{d}{dx}[g(x)] + g(x)\frac{d}{dx}[f(x)]$ **Example 2** Differentiate $y = x^2e^x$ **Solution** $\frac{dy}{dx} = x^2\frac{d}{dx}(e^x) + e^x\frac{d}{dx}(x^2)$ $= x^2e^x + e^x(2x)$ $= x^2e^x + 2xe^x$ ## Derivatives of Quotients **Example 3** Differentiate $y = \frac{x^2 + x - 2}{x^3 + 6}$ **Solution** $y' = \frac{(x^3 + 6)(2x + 1) - (x^2 + x - 2)(3x^2)}{(x^3 + 6)^2}$ $= \frac{2x^4 + x^3 + 12x + 6 - 3x^4 - 3x^3 + 6x^2}{(x^3 + 6)^2}$ $= \frac{-x^4 - 2x^3 + 6x^2 + 12x + 6}{(x^3 + 6)^2}$ ## Quotient Rule If $f$ and $g$ are differentiable, then $\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)\frac{d}{dx}[f(x)] - f(x)\frac{d}{dx}[g(x)]}{[g(x)]^2}$ **Example 4** Let $y = \frac{e^x}{1 + x^2}$. Find $y'$ **Solution** $y' = \frac{(1 + x^2)\frac{d}{dx}(e^x) - e^x\frac{d}{dx}(1 + x^2)}{(1 + x^2)^2}$ $= \frac{(1 + x^2)e^x - e^x(2x)}{(1 + x^2)^2}$ $= \frac{e^x + x^2e^x - 2xe^x}{(1 + x^2)^2}$ $= \frac{e^x(1 - 2x + x^2)}{(1 + x^2)^2}$ $= \frac{e^x(1 - x)^2}{(1 + x^2)^2}$