5 months ago
Q:

F(x,y)=3x-y/x+2y

Accepted Solution

A:

Solution variant #1.

Find the first partial derivative by treating the variable $y$ as a constant and differentiating with respect to $x$
$f_x=\frac{ \partial }{ \partial x } \left( \frac{ 3x-y }{ x+2y } \right)$
Use differentiation rule $\frac{ \partial }{ \partial x } \left( \frac{ f }{ g } \right)=\frac{ \frac{ \partial }{ \partial x } \left( f \right) \times g-f \times \frac{ \partial }{ \partial x } \left( g \right) }{ {g}^{2} }$
$f_x=\frac{ \frac{ \partial }{ \partial x } \left( 3x-y \right) \times \left( x+2y \right)-\left( 3x-y \right) \times \frac{ \partial }{ \partial x } \left( x+2y \right) }{ {\left( x+2y \right)}^{2} }$
Find the derivative of the sum or difference
$f_x=\frac{ 3\left( x+2y \right)-\left( 3x-y \right) \times \frac{ \partial }{ \partial x } \left( x+2y \right) }{ {\left( x+2y \right)}^{2} }$
Find the derivative of the sum or difference
$f_x=\frac{ 3\left( x+2y \right)-\left( 3x-y \right) \times 1 }{ {\left( x+2y \right)}^{2} }$
Simplify the expression
$f_x=\frac{ 7y }{ {\left( x+2y \right)}^{2} }$