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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

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

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

n = 0 \sum \infty (- \frac{1}{5})^{n}

\frac{d}{d x} (\frac{1}{x \frac{3}{2}})

(t e^{- t})^{^{'}}

\frac{\partial}{\partial x} (r cos (t))

\frac{d y}{d x} = \frac{1}{x} (y^{2} + 0.8 y), y (1) = 0.7

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