Practice section 3 of 8

Differentiate (a) $x\sin x$, (b) $e^{1+\cos x}$, and (c) $\ln\sqrt{1+\cos x}$ with respect to $x$.

Find $dy/dx$ in terms of $x$ and $y$ when (a) $x^2+y^3=xy^2$, and (b) $\cos x=1+y\cos y$.

Find $dy/dx$ given that $x+xy-2y=5$.

Find $dy/dx$ when (a) $y=x^{\sin x}$ for $x,y>0$, and (b) $x^y=y^x$ for $x,y>0$.

Prove by differentiation that $d(x^x)/dx=x^x(1+\ln x)$.

Differentiate (a) $\cos^{-1}(2x)$, (b) $\tan^{-1}(x^2-1)$, and (c) $\sin^{-1}(ax+b)$ with respect to $x$.