Question

Differentiate the following functions with respect to x :

$\frac{x^5-cosx}{sinx}$

Answer 1

y = $\frac{x^5-cosx}{sinx}$

$\frac{dy}{dx}=\frac{sinx\frac{d}{dx}(x^5-cosx)-(x^5-cosx)\frac{d}{dx}sinx}{(sinx{)}^2}$ [by quotient rule]

$=\frac{sinx(5x^4+sinx)-(x^5-cosx)}{si{n}^{2}x}=\frac{5x^{4}sinx+si{n}^{2}x-x^{5}cosx+co{s}^{2}x}{si{n}^{2}x}$

$=\frac{5x^{4}sinx-x^{5}cosx+si{n}^{2}x+co{s}^{2}x}{si{n}^{2}x}=\frac{5x^{4}sinx-x^{5}cosx+1}{si{n}^{2}x}$