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

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

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

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

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

Solution steps

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

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

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

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

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

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

y^{^{'''}} - 2 y^{^{''}} - y^{^{'}} + 2 y = 0

3 r = \frac{d r}{d θ} - θ^{3}

\int cos (\frac{1}{5} x) d x

\int sin (x) + 1 d x

\frac{d}{d x} (ln (c) os x)

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