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

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

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

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

\frac{y ^{2}}{y}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{y ^{2}}{y}

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

\frac{d}{d x} (- cos (x))

x \to - 1 lim (- x^{2} + 2 x - 9)

- y^{^{'}} + \frac{4 y}{x} = x y^{2}

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

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