Question

If $y=a^{\frac{1}{2}{\log }_a\cos x}$, find $\frac{dy}{dx}$

Answer 1

Given $y=a^{\frac{1}{2}{\log }_a\cos x}$

${\log }_a{b}^m=m{\log }_{a}b$

$y=a^{{\log }_a{\left(\cos x\right)}^{\frac{1}{2}}}$

$y={\left(\cos x\right)}^{\frac{1}{2}}$

$y^2=\cos x$

Differentiating wrt $x$

$2y{y}^{\prime }=-\sin x$

$y^{\prime }=-\frac{\sin x}{2\sqrt{\cos x}}$