What is (\partial)/(\partial x)(x/(x^3+y^2)) ?

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

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(\partial)/(\partial x)(x/(x^3+y^2))

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

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

Solution steps

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

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

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

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

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

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

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

\frac{d}{d x} (lo g_{10} (x^{6} + 3))

\int \frac{1}{y + 1} d y

\frac{sin ( t )}{4 + 4 cos ( t )}

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

\int_{0}^{\frac{π}{2}} (sin (x))^{2} d x

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