91Ó°ÊÓ

\begin{equation}f(x, y, z)=x^2 y^3 \sqrt{z}\end{equation}

The Rate of change of f in the z-direction(positive) when the values of x and y are constant is partially differentiating the given function f with respect to z i.e

\begin{equation} \frac{\partial f}{\partial z}=x^{2}y^{^{3}}\frac{\partial }{\partial z}\left ( z^{1/2} \right )\end{equation}

\begin{equation}\frac{\partial f}{\partial z}=x^{2}y^{^{3}}\left ( 1/2 \right )\left ( z^{1/2-1} \right )\end{equation}

\begin{equation} \frac{\partial f}{\partial z}=\frac{x^{2}y{^{3}}}{2}\left ( z^{-1/2} \right)\end{equation}

\begin{equation}\frac{\partial f}{\partial z}=\frac{x^{2}y{^{3}}}{2\sqrt{z}}\end{equation}

The Rate of change of f in the y-direction(positive) when the values of x and z are constant is partially differentiating the given function f with respect to y i.e

\begin{equation}\frac{\partial f}{\partial y}=x^{^{2}}\left ( \frac{\partial }{\partial y}y^{3}\right)\sqrt{z}\end{equation}

\begin{equation}\frac{\partial f}{\partial y}=x^{^{2}}(3*(y^{^{3-1}})\sqrt{z}\end{equation}

\begin{equation}\frac{\partial f}{\partial y}=x^{^{2}}(3y^{^{2}})\sqrt{z}\end{equation}

\begin{equation}\frac{\partial f}{\partial y}=3x^{^{2}}y^{^{2}}\sqrt{z}\end{equation}

The Rate of change of f in the x-direction(positive) when the values of y and z are constant is partially differentiating the given function f with respect to x i.e

\begin{equation}\frac{\partial f}{\partial x}=\left ( \frac{\partial }{\partial x}(x^{2}) \right )y^{^{3}}\sqrt{z}\end{equation}

\begin{equation}\frac{\partial f}{\partial x}=(2x^{2-1})y^{^{3}}\sqrt{z}\end{equation}

\begin{equation}\frac{\partial f}{\partial x}=2xy^{^{3}}\sqrt{z}\end{equation}