What is (\partial)/(\partial x)((e^y)/(x+y^3)) ?

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

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(\partial)/(\partial x)((e^y)/(x+y^3))

\frac{\partial}{\partial x} (\frac{e ^{y}}{x + y ^{3}})

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

Solution steps

\frac{\partial}{\partial x} (\frac{e ^{y}}{x + y ^{3}})

Treat

y

as a constant

Take the constant out:

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

= e^{y} \frac{\partial}{\partial x} (\frac{1}{x + y ^{3}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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