Section 2.3 Products and Quotients of Functions
Let \(f(x) = 3x^2\) and \(g(x) = 4x^3\text{.}\) Find \(f'(x)\text{,}\) \(g'(x)\text{,}\) and \(\dfrac{d}{dx} \left( f(x) \cdot g(x) \right)\text{:}\)
\begin{equation*}
f'(x) = \dfrac{d}{dx} (3x^2) = 6x
\end{equation*}
\begin{equation*}
g'(x) = \dfrac{d}{dx} (4x^3) = 12x^2
\end{equation*}
\begin{align*}
\dfrac{d}{dx} \left( f(x) \cdot g(x) \right) \amp = \dfrac{d}{dx}\left( (3x^2) (4x^3) \right)\\
\amp = \dfrac{d}{dx}(12x^5)\\
\amp = 60x^4
\end{align*}
Theorem 2.3.1.
Suppose \(f\) and \(g\) are differentiable; then, so is the product \(f \cdot g\) and \(\dfrac{d}{dx}\left( f(x) \cdot g(x) \right) = f(x) \cdot \dfrac{dg}{dx} + g(x) \cdot \dfrac{df}{dx}\)
Example 2.3.1.
Find \(f'(x)\) for \(f(x) = \sqrt{x}(3x^2 + 7)\)
\begin{align*}
f'(x) \amp = \dfrac{d}{dx} \left( \sqrt{x}(3x^2 + 7) \right) \\
\amp = \sqrt{x} \cdot \dfrac{d}{dx}(3x^2 + 7) + (3x^2 + 7) \cdot \dfrac{d}{dx}(\sqrt{x}) \\
\amp = \sqrt{x} (6x) + (3x^2 + 7)(\dfrac{1}{2}x^{-\frac{1}{2}})\\
\amp = \sqrt{x} (6x) + \dfrac{3x^2 + 7}{2\sqrt{x}}
\end{align*}
Theorem 2.3.2. Quotient Rule.
Suppose \(f\) and \(g\) are differentiable and \(g(x) \neq 0\text{,}\) \(\dfrac{d}{dx} \left( \dfrac{f(x)}{g(x)} \right) = \dfrac{g(x) \cdot \frac{df}{dx} - f(x) \cdot \frac{dg}{dx}}{(g(x))^2}\)
Example 2.3.2.
Find \(\dfrac{d}{dx} \left( \dfrac{1}{x^5} \right)\text{:}\)
\begin{align*}
\dfrac{d}{dx}\left( \dfrac{1}{x^5} \right) \amp = \dfrac{x^5 \cdot \frac{d}{dx}(1) - 1 \cdot \frac{d}{dx}(x^5)}{(x^5)^2}\\
\amp = \dfrac{x^5 (0) - 1 (5x^4)}{x^{10}}\\
\amp = -\dfrac{5x^4}{x^{10}}\\
\amp = -\dfrac{5}{x^6}
\end{align*}