# Find all first partial derivatives, and evaluate each at the given point. f(x,y)=x^2-y,(0,2)

Find all first partial derivatives, and evaluate each at the given point. $f\left(x,y\right)={x}^{2}-y,\left(0,2\right)$
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Theodore Schwartz
Step 1
The given function is $f\left(x,y\right)={x}^{2}-y$ and a point is (0, 2).
Compute the partial derivatives as follows.
${f}_{x}\left(x,y\right)=\frac{\partial }{\partial x}\left({x}^{2}-y\right)$
$=2x-0$
$=2x$
${f}_{y}\left(x,y\right)=\frac{\partial }{\partial y}\left({x}^{2}-y\right)$
$=0-1=-1$
Thus, the first partial derivatives are ${f}_{x}=2x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{y}=-1$.
Step 2
Substitute $\left(x,y\right)=\left(0,2\right)\in {f}_{x}=2x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{y}=-1$ and obtain that,
${f}_{x}\left(0,2\right)=0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{y}\left(0,2\right)=-1$