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

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

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

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

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

Solution steps

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

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

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

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

Simplify

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

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

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