Question

If ${\left(\cos x\right)}^y={\left(\cos y\right)}^x$ then find $\frac{dy}{dx}$

Answer 1

$(\cos x{)}^y=(\cos y{)}^x\log (\cos x{)}^y=\log (\cos y{)}^{x}y\log \left(\cos x\right)=x\log \left(\cos y\right)$

Differentiating w.r.t $x$

$\frac{y}{\cos x}+\cos x\frac{dy}{dx}=\cos y-x\sin y\frac{dy}{dx}\frac{dy}{dx}(\cos x+x\sin x)=\cos y-\frac{y}{\cos x}\frac{dy}{dx}=\frac{\cos y-\frac{y}{\cos x}}{\cos x+x\sin x}$