Find the differential of each of the following expressions; zap each of the following with \(d\). Note that all italicized letters are variables:
\[f=3x-5z^2+2xy\]
\begin{equation*} df = 3\ dx - 10 z\ dz + 2x\ dy + 2y\ dx \end{equation*}
\[g=\frac{c^{1/2}b}{a^2}\]
\begin{equation*} dg=\frac{c^{-1/2}b}{2a^2} dc + \frac{c^{1/2}}{a^2} db -\frac{2c^{1/2}b}{a^3} da \end{equation*}
\[h=\sin^2(\omega t)\]
\begin{equation*} dh = 2 \sin(\omega t) \cos(\omega t) (\omega\ dt + t\ d\omega) = \sin(2 \omega t) (\omega\ dt + t\ d\omega) \end{equation*}
\[j=a^x\]
\begin{equation*} dj = a^x \ln(a)\ dx + x a^{x-1}da \end{equation*}
\[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]
\begin{equation*} dk = 5\sec^2\left(\ln\left(\frac{V_1}{V_2}\right)\right) \left(\frac{dV_1}{V_1} - \frac{dV_2}{V_2}\right) \end{equation*}