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

The answer to (\partial)/(\partial x)((y-x^2)(2y^2+x)) is -3x^2-4y^2x+y

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

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

- 3 x^{2} - 4 y^{2} x + y

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

(- 2 x) (2 y^{2} + x) + 1 \cdot (y - x^{2}) : - 3 x^{2} - 4 y^{2} x + y

= - 3 x^{2} - 4 y^{2} x + y

\int \frac{5}{36 + x ^{2}} d x

\frac{\partial}{\partial y} (x y (1 - x - y))

y = x^{4} + 2 e^{x}

x \to 9 lim (∣ x + 3 ∣)

3 t \cdot \frac{d y}{d t} - y = t

What is (\partial)/(\partial x)((y-x^2)(2y^2+x)) ?
The answer to (\partial)/(\partial x)((y-x^2)(2y^2+x)) is -3x^2-4y^2x+y