Best solutions to - "The following questions are about the function f(x,y)=x2e2xy...."

Mylo O'Moore

Answered question

2021-10-08

The following questions are about the function

f (x, y) = x^{2} e^{2 x y}

.
Find the partial derivatives,

f_{x} (x, y) and f_{y} (x, y)

Arnold Odonnell

Skilled2021-10-09Added 109 answers

Step 1
If f(x,y) is a function of two variables x and y, then

f_{x} (x, y) = \frac{\partial f}{\partial x}

f_{y} (x, y) = \frac{\partial f}{\partial y}

The equation of the tangent plane of function f at the point

(x_{0}, y_{0})

z = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})

Step 2
The given function is

f (x, y) = x^{2} e^{2 x y}

Differentiate the given function partially with respect to x.

f_{x} (x, y) = x^{2} \frac{\partial}{\partial x} (e^{2 x y}) + e^{2 x y} \frac{\partial}{\partial x} x^{2}

= x^{2} (2 y e^{2 x y}) + e^{2 x y} (2 x)

= 2 x e^{2 x y} (x y + 1)

Differentiate the given function partially with respect to y.

f_{y} (x, y) = x^{2} \frac{\partial}{\partial y} (e^{2 x y})

= x^{2} (2 x e^{2 x y})

= 2 x^{3} e^{2 x y}

Jeffrey Jordon

Expert2022-08-15Added 2605 answers

Answer is given below (on video)

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