Products and Quotients

Basic Concept
Video

Products and Quotients

$f (x)$	$f^{'} (x)$
$y = u . v$	$y^{'} = u^{'} . v + u . v^{'}$
$y = u . v . w$	$y^{'} = u^{'} . v . w + u . v^{'} . w + u . v . w^{'}$
$y = \frac{u}{v}$	$y^{'} = \frac{u^{'} . v - u . v^{'}}{v^{2}}$

Example 01

$y = (x^{2} + 3 x + 4) . (6 x + 2)$

$\begin{aligned} u & = x^{2} + 3 x + 4 \to u^{'} = 2 x + 3 \\ v & = 6 x + 2 \to v^{'} = 6 \\ y^{'} & = u^{'} . v + u . v^{'} \\ y^{'} & = (2 x + 3) . (6 x + 2) + (x^{2} + 3 x + 4) . 6 \\ y^{'} & = 12 x^{2} + 18 x + 4 x + 6 + 6 x^{2} + 18 x + 24 \\ y^{'} & = 18 x^{2} + 40 x + 30 \end{aligned}$

Example 02

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

$\begin{aligned} u & = 2 x \to u^{'} = 2 \\ v & = 3 x + 1 \to v^{'} = 3 \\ y^{'} & = \frac{u^{'} . v - u . v^{'}}{v^{2}} \\ y^{'} & = \frac{2 . (3 x + 1) - 2 x . 3}{(3 x + 1)^{2}} \\ y^{'} & = \frac{6 x + 2 - 6 x}{(3 x + 1)^{2}} \\ y^{'} & = \frac{2}{(3 x + 1)^{2}} \end{aligned}$

Exercise

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Products and Quotients

Exercise

Differentiation