Differentiate the following functions with respect to x:
$(1 - 2 t a n x) (5 + 4 s i n x)$

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Solution

We have,
Differentiate the following functions with respect to x:
$\frac{d}{d x} {(1 - 2 t a n x) (5 + 4 s i n x)}$
$= (5 + 4 s i n x) \frac{d}{d x} (1 - 2 t a n x) + (1 - 2 t a n x) \frac{d}{d x} (5 + 4 s i n x)$ [Using product rule]
$= (5 + 4 s i n x) (0 - 2 s e c^{2} x) + (1 - 2 t a n x) (0 + 4 c o s x)$
$= - 10 s e c^{2} x - 8 s i n x \times s e c^{2} x + 4 c o s x - 8 c o s x \times t a n x$
$= 4 (\frac{- 5}{2} s e c^{2} x - 2 s i n x \times \frac{1}{c o s^{2} x} + c o s x - 2 c o s x \times \frac{s i n x}{c o s x})$
$= 4 (\frac{- 5}{2} s e c^{2} x - 2 t a n x s e c x + c o s x - 2 s i n x)$
$= 4 (c o s x - 2 s i n x - 2 t a n x s e c x - \frac{5}{2} s e c^{2} x)$

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