Practice section 2 of 8

Differentiate (a) $2/\sqrt{x}+6/\sqrt[3]{x}$ and (b) $(x^2+3)^4(2x^3-5)^3$ with respect to $x$.

Differentiate (a) $x^2\sin x$, (b) $(1+\tan x)/\sec x$, (c) $\sqrt{\cos 2x}$, and (d) $\cot(x^2)+\sec^2x$ with respect to $x$.

Prove that $d(\cot x)/dx=-\csc^2x$.

Given $y=\frac32x+2\sin x+\frac12\sin x\cos x$, show that $dy/dx=(1+\cos x)^2$.

Differentiate (a) $4e^{x^2+1}$, (b) $3^{\tan x}$, (c) $\ln(x+x^3)$, and (d) $\log_3(x^2+4)$ with respect to $x$.

Differentiate $\ln\sqrt{e^x/(2+3^{x^2})}$ with respect to $x$.