Find the differentiation of f(x) = \(\rm \frac{(a\cos x + b \sin x + c)}{\sin x}\)

Concept:

\(\rm \frac{\mathrm{d} (cotx)}{\mathrm{d} x} = -cosec^{2}x \)

\(\rm \frac{\mathrm{d} (cosecx)}{\mathrm{d} x} = - cotx\; cosecx\)

Calculation:

Given: f(x) = \(\rm \frac{(a\cos x + b \sin x + c)}{\sin x}\) = acot x + b + c cosec x

Now, f'(x) = \(\rm \frac{\mathrm{d} (a\cot x + b + c(cosec\; x))}{\mathrm{d} x}\)

⇒ -acosec²x – (c × cosec x cot x)

∴ The required value is -acosec2x – (c × cosec x cot x)