Differentiate \(\frac{x}{sinx}\) with respect to x.

1.	Differentiate \(\frac{x}{sinx}\) with respect to x.
Answer» Let f(x) = \(\frac{x}{sinx}\) ∴ f(x) = \(\frac{d}{dx}(\frac{x}{sinx})\) = \(\frac{sinx\frac{d}{dx}x-x\frac{d}{dx}sinx}{(sinx)^2}\) = \(\frac{sinx.1-x.cosx}{sin^2x}\) = \(cosec\,x-x\,cot\,x\,cosec\,x\) = \((1-x\,cot\,x).cosec\,x\)

Answer»

Let f(x) = \(\frac{x}{sinx}\)

∴ f(x) = \(\frac{d}{dx}(\frac{x}{sinx})\)

= \(\frac{sinx\frac{d}{dx}x-x\frac{d}{dx}sinx}{(sinx)^2}\)

= \(\frac{sinx.1-x.cosx}{sin^2x}\)

= \(cosec\,x-x\,cot\,x\,cosec\,x\)

= \((1-x\,cot\,x).cosec\,x\)

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