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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{1}{3 x ^{2} + 5 x})

\int e^{- y^{2}} y d y

y^{^{'}} = 5 - y

\int_{0}^{5} \frac{x}{1 + 4 x} d x

\int \frac{2 x ^{3} - 3 x ^{2} - 3 x + 2}{x - 2} d x

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