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

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

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

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

3 x^{2} - 2 x y^{2} + y

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2} + y, g = x - y^{2}

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

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

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

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

Simplify

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

= 3 x^{2} - 2 x y^{2} + y

\int x^{31} sin (x^{32}) d x

x y^{^{'}} + y = 46 x

e^{(- 3 x)}

y = \frac{3 x ^{4} + 9}{x ^{2}}

\frac{x ^{3} - 2 x ^{2} + 4}{x ^{2}}

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