What is (\partial)/(\partial x)(((-x+y))/(x*y+3*x)) ?

The answer to (\partial)/(\partial x)(((-x+y))/(x*y+3*x)) is -y/(x^2(y+3))

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(\partial)/(\partial x)(((-x+y))/(xy+3x))

\frac{\partial}{\partial x} (\frac{( - x + y )}{x \cdot y + 3 \cdot x})

- \frac{y}{x ^{2} ( y + 3 )}

Solution steps

\frac{\partial}{\partial x} (\frac{( - x + y )}{x y + 3 x})

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial x} ( - x + y ) ( x y + 3 x ) - \frac{\partial}{\partial x} ( x y + 3 x ) ( - x + y )}{( x y + 3 x ) ^{2}}

\frac{\partial}{\partial x} (- x + y) = - 1

\frac{\partial}{\partial x} (x y + 3 x) = y + 3

= \frac{( - 1 ) ( x y + 3 x ) - ( y + 3 ) ( - x + y )}{( x y + 3 x ) ^{2}}

Simplify

\frac{( - 1 ) ( x y + 3 x ) - ( y + 3 ) ( - x + y )}{( x y + 3 x ) ^{2}} : - \frac{y}{x ^{2} ( y + 3 )}

= - \frac{y}{x ^{2} ( y + 3 )}

y^{^{'}} = y^{2} (5 + x)

\frac{\partial}{\partial x} (\frac{z}{1 + z ^{2} y ^{2}})

\frac{d}{d x} (A x e^{3 x})

f (t) = sin (t) - t cos (t)

f^{^{'}} (x) = \frac{1 - e ^{x}}{9 + e ^{x}}

What is (\partial)/(\partial x)(((-x+y))/(x*y+3*x)) ?
The answer to (\partial)/(\partial x)(((-x+y))/(x*y+3*x)) is -y/(x^2(y+3))