Given f(x, y) = 3x°– læ?y? – 5y³, find-fa(x, y)fy(x, y) =fæx (x, y) =fay (x, y) =

Answered: Given f(x, y) = 3x® – læ²y² – 5y³, find…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Given f(x, y) = 3x°
– læ?y? – 5y³, find
-
fa(x, y)
fy(x, y) =
fæx (x, y) =
fay (x, y) =

Expert Solution

Step 1

Given that, the function is $f (x, y) = 3 x^{6} - 1 x^{2} y^{2} - 5 y^{3}$ .

Obtain the value of $f_{x} (x, y)$ .

$\begin{array}{rcl} f_{x} (x, y) & = & \frac{\partial}{\partial x} (3 x^{6} - 1 x^{2} y^{2} - 5 y^{3}) \\ = & \frac{\partial}{\partial x} (3 x^{6}) - y^{2} \frac{\partial}{\partial x} (x^{2}) - 5 y^{3} \frac{\partial}{\partial x} (1) \\ = & 18 x^{5} - 2 x y^{2} \end{array}$

Obtain the value of $f_{y} (x, y)$ .

$\begin{array}{rcl} f_{y} (x, y) & = & \frac{\partial}{\partial y} (3 x^{6} - 1 x^{2} y^{2} - 5 y^{3}) \\ = & 3 x^{6} \frac{\partial}{\partial y} (1) - x^{2} \frac{\partial}{\partial y} (y^{2}) - 5 \frac{\partial}{\partial y} (y^{3}) \\ = & 0 - 2 x^{2} y - 15 y^{2} \\ = & - 2 x^{2} - 15 y^{2} \end{array}$

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