What is (\partial)/(\partial x)(x*(e^y)/(x+y)) ?

The answer to (\partial)/(\partial x)(x*(e^y)/(x+y)) is (e^yy)/((x+y)^2)

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(\partial)/(\partial x)(x*(e^y)/(x+y))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = \frac{1}{x + y}

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

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

n = 8 \sum \infty \frac{n}{n ln ( n )}

\frac{\partial}{\partial x} (ln (x + 8 y + 9 z))

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

x \to 1 - lim (\frac{x}{x + 1})

\frac{d}{d x} (\frac{2}{5 x ^{2} - 1})

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