Question

. -) Obtain derivaties of the following functions x/sin x.
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Answer 1

Answer:

your solution is as follows

pls mark it as brainliest

the above pics contains proof by first principle

while the written answer consists of u/v rule

Step-by-step explanation:

$to \: find : \frac{d}{dx} ( \frac{x}{sin \: x} \: ) \\ \\ we \: know \: that \\ according \: to \: quotient \: rule \\ \frac{d}{dx}( \frac{u}{v} ) = \frac{v. \frac{d}{dx} (u) - u. \frac{d}{dx}(v) }{v {} } \\ \\ = \frac{sin \: x. \frac{d}{dx} (x) - x. \frac{d}{dx} (sin \: x)}{sin {}^{2} x} \\ \\ = \frac{sin \: x(1) - x.cos \: x}{sin {}^{2}x } \\ \\ = \frac{sin \: x}{sin {}^{2} \: x} - \frac{x.cos \: x}{sin {}^{2} x} \\ \\ = \frac{1}{sin \: x} - \frac{x.cos \: x}{sin \: x} . \frac{1}{sin \: x} \\ \\ = cosec \: x - x.cot \: x.cosec \: x \\ \\ = cosec \: x(1 - x.cot \: x)$