Mylo O'Moore

2021-10-08

The following questions are about the function $f\left(x,y\right)={x}^{2}{e}^{2xy}$.
Find the partial derivatives, .

Arnold Odonnell

Step 1
If f(x,y) is a function of two variables x and y, then
${f}_{x}\left(x,y\right)=\frac{\partial f}{\partial x}$
${f}_{y}\left(x,y\right)=\frac{\partial f}{\partial y}$
The equation of the tangent plane of function f at the point $\left({x}_{0},{y}_{0}\right)$ is
$z=f\left({x}_{0},{y}_{0}\right)+{f}_{x}\left({x}_{0},{y}_{0}\right)\left(x-{x}_{0}\right)+{f}_{y}\left({x}_{0},{y}_{0}\right)\left(y-{y}_{0}\right)$
Step 2
The given function is
$f\left(x,y\right)={x}^{2}{e}^{2xy}$
Differentiate the given function partially with respect to x.
${f}_{x}\left(x,y\right)={x}^{2}\frac{\partial }{\partial x}\left({e}^{2xy}\right)+{e}^{2xy}\frac{\partial }{\partial x}{x}^{2}$
$={x}^{2}\left(2y{e}^{2xy}\right)+{e}^{2xy}\left(2x\right)$
$=2x{e}^{2xy}\left(xy+1\right)$
Differentiate the given function partially with respect to y.
${f}_{y}\left(x,y\right)={x}^{2}\frac{\partial }{\partial y}\left({e}^{2xy}\right)$
$={x}^{2}\left(2x{e}^{2xy}\right)$
$=2{x}^{3}{e}^{2xy}$

Jeffrey Jordon

Answer is given below (on video)

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