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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

\int \frac{4}{sec ( 2 x ) - 1} d x

\int_{0}^{3} \frac{x - 1}{x ^{2} - 3 x - 4} d x

\int \frac{7 x ^{3} + 4 x + 4 x ^{2} + 1}{x ^{4} + x ^{2}} d x

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

\int \frac{4 x}{x - 6} d x

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