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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

f = y^{2}, g = e^{x y^{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 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 e^{y^{2} x} y + 2 e^{y^{2} x} y^{3} x

\int z sin (x) d y

\frac{1}{3} x^{3} + \frac{7}{2} x^{2} - 3 x + 5

f (t) = tan (t)

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

\int sin^{12} (x) d x

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