Question

Differentiate the function given below w.r.t. $x:$

$\frac{x^5-\cos x}{\sin x}$

Answer 1

Let $y=\frac{x^5-\cos x}{\sin x}$

$\Rightarrow \frac{dy}{dx}=\frac{\left(\begin{array}{c}\sin x\frac{d(x^5-\cos x)}{dx}-\\ (x^5-\cos x)\frac{d\left(\sin x\right)}{dx}\end{array}\right)}{(\sin x{)}^2}$

$\Rightarrow \frac{dy}{dx}=\frac{\left(\begin{array}{c}\sin x[\frac{d\left(x^5\right)}{dx}-\frac{d\left(\cos x\right)}{dx}]-\\ (x^5-\cos x)\frac{d\left(\sin x\right)}{dx}\end{array}\right)}{(\sin x{)}^2}$

$\Rightarrow \frac{dy}{dx}=\frac{\sin x(5x^4+\sin x)-(x^5-\cos x)\cos x}{(\sin x{)}^2}$

$\Rightarrow \frac{dy}{dx}=\frac{5x^4\sin x-x^5\cos x+{\sin }^{2}x+{\cos }^{2}x}{(\sin x{)}^2}$

$\Rightarrow \frac{dy}{dx}=\frac{5x^4\sin x-x^5\cos x+1}{{\sin }^{2}x}$