What is (\partial)/(\partial x)(x/(x+y)+e^{xy}x+4y) ?

The answer to (\partial)/(\partial x)(x/(x+y)+e^{xy}x+4y) is e^{xy}xy+e^{xy}+y/((x+y)^2)

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(\partial)/(\partial x)(x/(x+y)+e^{xy}x+4y)

\frac{\partial}{\partial x} (\frac{x}{x + y} + e^{x y} x + 4 y)

e^{x y} x y + e^{x y} + \frac{y}{( x + y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{x}{x + y} + e^{x y} x + 4 y)

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (\frac{x}{x + y}) + \frac{\partial}{\partial x} (e^{x y} x) + \frac{\partial}{\partial x} (4 y)

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

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

\frac{\partial}{\partial x} (4 y) = 0

= \frac{y}{( x + y ) ^{2}} + y e^{y x} x + e^{y x} + 0

Simplify

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

\frac{d}{d x} (\frac{x - x ^{2}}{x})

\frac{d}{d x} (2 cos (\frac{x}{2}))

- 4 (x - 9)^{3}

(sin (x)), \frac{π}{2}

n = 1 \sum \infty \frac{2 n}{3 n}

What is (\partial)/(\partial x)(x/(x+y)+e^{xy}x+4y) ?
The answer to (\partial)/(\partial x)(x/(x+y)+e^{xy}x+4y) is e^{xy}xy+e^{xy}+y/((x+y)^2)