Question

If $y=x\left[\right(\cos \frac{x}{2}+\sin \frac{x}{2}\left)\right(\cos \frac{x}{2}-\sin \frac{x}{2})+\sin x]+\frac{1}{2\sqrt{x}}$ find $\frac{dy}{dx}$

Answer 1

$y=x[(\cos \frac{x}{2}+\sin \frac{x}{2})(\cos \frac{x}{2}-\sin \frac{x}{2})+\sin x]+\frac{1}{2\sqrt{x}}$

$=x\{[{\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}]+\sin x\}+\frac{1}{2\sqrt{x}}[\because (a+b\left)\right(a-b)=a^2-b^2]$

$=x(\cos x+\sin x)+\frac{1}{2\sqrt{x}}$

$\therefore \frac{dy}{dx}=\frac{d}{dx}\left(x\right).[\cos x+\sin x]+x\left\{\frac{d}{dx}\right(\cos x+\sin x\left)\right\}+\frac{1}{2}\frac{d}{dx}\left(x^{-1/2}\right)$

$\therefore \frac{dy}{dx}=(\cos x+\sin x)+x(\cos x-\sin x)-\frac{1}{4}{x}^{-3/2}$