Find the derivative of tan ⁡ ( π x ) sin ⁡ ( π x ).

Question

Find the derivative of   tan  ⁡  (  π  x  )      ﻿    sin  ⁡  (  π  x  ).

candydulce168nlid · Accepted Answer

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \tan (π x)$ and $g (x) = \sin (π x)$ .

$\tan (π x) \frac{d}{d x} [\sin (π x)] + \sin (π x) \frac{d}{d x} [\tan (π x)]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = π x$ .

$\tan (π x) (\cos (π x) \frac{d}{d x} [π x]) + \sin (π x) \frac{d}{d x} [\tan (π x)]$

Differentiate.

$\tan (π x) \cos (π x) π + \sin (π x) \frac{d}{d x} [\tan (π x)]$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = π x$ .

$\tan (π x) \cos (π x) π + \sin (π x) (\sec^{2} (π x) \frac{d}{d x} [π x])$

Differentiate.

$\tan (π x) \cos (π x) π + \sin (π x) \sec^{2} (π x) π$

Simplify.

$π \sin (π x) + π s e c (π x) \tan (π x)$

Find the derivative of tan ⁡<!-- ⁡

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