Partial Derivitive

If a function has more than one variable, say {f(x,y,z)}, we are able to take a partial derivative of the function in respect to one of its variable:

    \begin{equation*} {\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\normalsubformula{\text{etc}}} \end{equation*}

Iterated as:

    \begin{equation*} {\frac{\partial f}{\partial x}=\underset{{h\rightarrow 0}}{{\text{lim}}}\frac{f(x+h,y)-f(x,y)}{h}} \end{equation*}

    \begin{equation*}{\frac{\partial f}{\partial y}=\underset{{h\rightarrow 0}}{{\text{lim}}}\frac{f(x,y+k)-f(x,y)}{k}}\end{equation*}

, etc

Example

Find the first order partial derivatives for the function:

    \begin{equation*} {f(x,y)=x^{{5}}+4\sqrt{y}-\text{10}} \end{equation*}

First take the derivate with respect to x treating all coefficients with a y as a constant.   The partial derivative with respect to x is:

    \begin{equation*} {\frac{\partial f}{\partial x}=5x^{{4}}} \end{equation*}

Take note that the second and third terms in the equation differentiate to zero when the partial derivate with respectto x is taken, so they are eliminated.

The next step is to take the partial derivative with respect to y while treating all the coefficients with an x as a constant.   The partial derivative with respect to y is:

    \begin{equation*} {\frac{\partial f}{\partial y}=\frac{2}{\sqrt{y}}} \end{equation*}