Differentiation

Products and Quotients

Products and Quotients
 $$f(x)$$ $$f'(x)$$ $$y = u \:.\: v$$ $$y' = u' \:.\: v + u \:.\: v'$$ $$y = u \:.\: v \:.\: w$$ $$y' = u' \:.\: v \:.\: w + u \:.\: v' \:.\: w + u \:.\: v \:.\: w'$$ $$y = \dfrac uv$$ $$y' = \dfrac {u' \:.\: v - u \:.\: v'}{v^2}$$

Example 01

$$y = (x^2 + 3x + 4) \:.\: (6x + 2)$$

\begin{equation*} \begin{split} u & = x^2 + 3x + 4 \rightarrow u' = 2x + 3 \\\\ v & = 6x + 2 \rightarrow v' = 6 \\\\\\ y' & = u' \:.\: v + u \:.\: v' \\\\ y' & = ( 2x + 3) \:.\: (6x + 2) + (x^2 + 3x + 4) \:.\: 6 \\\\ y' & = 12x^2 + 18x + 4x + 6 + 6x^2 + 18x + 24 \\\\ y' & = \bbox[5px, border: 2px solid magenta] {18x^2 + 40x + 30} \end{split} \end{equation*}

Example 02

$$y = \dfrac{2x}{3x + 1}$$

\begin{equation*} \begin{split} u & = 2x \rightarrow u' = 2 \\\\ v & = 3x + 1 \rightarrow v' = 3 \\\\\\ y' & = \frac {u' \:.\: v - u \:.\: v'}{v^2} \\\\ y' & = \frac {2 \:.\: (3x + 1) - 2x \:.\: 3}{(3x + 1)^2} \\\\ y' & = \frac {6x + 2 - 6x}{(3x + 1)^2} \\\\ y' & = \bbox[5px, border: 2px solid magenta] {\frac {2}{(3x + 1)^2}} \end{split} \end{equation*}

Exercise

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