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Algebra of Derivatives

Differentiate...

Question

Differentiate $- \frac{4 x + 5 s i n x}{3 x + 7 c o s x}$ w.r.t $x$

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Solution

$\frac{d}{d x} (\frac{- 4 x + 5 s i n x}{3 x + 7 c o s x}) = - \frac{d}{d x} (\frac{4 x + 5 s i n x}{3 x + 7 c o s x})$
Now $(\frac{U}{V}) = \frac{V U^{1} - U V^{1}}{V^{2}}$
$∴= - [\frac{(3 x + 7 c o s x) \frac{d}{d x} (4 x + 5 s i n x) - (4 x + 5 s i n x) \frac{d}{d x} (3 x + 7 c o s x)}{(3 x + 7 c o s x)^{2}}]$
$= - [\frac{(3 x + 7 c o s x) (4 + 5 c o s x) - (4 x + 5 s i n x) (3 - 7 s i n x)}{(3 x + 7 c o s x)^{2}}]$
$= - [\frac{12 x + 15 x c o s x + 28 c o s x + 35 c o s^{2} x - 12 x + 28 x s i n x - 15 s i n x + 35 s i n^{2} x}{(3 x + 7 c o s x)^{2}}]$
$= - [\frac{15 x c o s x + 28 x s i n x + 28 c o s x - 15 s i n x + 35}{(3 x + 7 c o s x)^{2}}]$
$= - [\frac{15 (x c o s x - s i n x) + 28 (x s i n x + c o s x) + 35}{(3 x + 7 c o s x)^{2}}]$

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