What is (\partial)/(\partial y)(2y^2e^{xy^2}) ?

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

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(\partial)/(\partial y)(2y^2e^{xy^2})

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

2 (2 e^{y^{2} x} y + 2 e^{y^{2} x} y^{3} x)

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = y^{2}, g = e^{x y^{2}}

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

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

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

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

e^{x y^{2}} \cdot 2 x y y^{2} = 2 e^{y^{2} x} y^{3} x

= 2 (2 e^{y^{2} x} y + 2 e^{y^{2} x} y^{3} x)

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

7 cos (x)

\int_{- \infty}^{0} x e^{- 8 x} d x

(1 + 2 x)^{2}

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