What is (\partial)/(\partial y)(xsqrt(1+y^3)) ?

The answer to (\partial)/(\partial y)(xsqrt(1+y^3)) is (3xy^2)/(2sqrt(1+y^3))

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(\partial)/(\partial y)(xsqrt(1+y^3))

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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\frac{\partial}{\partial y} (5 x^{2} + 2 y^{2})

\int 13 cos^{3} (x) d x

\frac{d P}{d t} = \frac{P}{5} - \frac{P ^{2}}{75}, P (0) = 5

What is (\partial)/(\partial y)(xsqrt(1+y^3)) ?
The answer to (\partial)/(\partial y)(xsqrt(1+y^3)) is (3xy^2)/(2sqrt(1+y^3))