What is (\partial)/(\partial x)(xy^2e^{3x^2y-y^2}) ?

The answer to (\partial)/(\partial x)(xy^2e^{3x^2y-y^2}) is y^2(e^{3x^2y-y^2}+6e^{3x^2y-y^2}x^2y)

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(\partial)/(\partial x)(xy^2e^{3x^2y-y^2})

\frac{\partial}{\partial x} (x y^{2} e^{3 x^{2} y - y^{2}})

y^{2} (e^{3 x^{2} y - y^{2}} + 6 e^{3 x^{2} y - y^{2}} x^{2} y)

Solution steps

\frac{\partial}{\partial x} (x y^{2} e^{3 x^{2} y - y^{2}})

Treat

y

as a constant

Take the constant out:

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

= y^{2} \frac{\partial}{\partial x} (x e^{3 x^{2} y - y^{2}})

Apply the Product Rule:

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

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

= y^{2} (\frac{\partial}{\partial x} (x) e^{3 x^{2} y - y^{2}} + \frac{\partial}{\partial x} (e^{3 x^{2} y - y^{2}}) x)

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (e^{3 x^{2} y - y^{2}}) = e^{3 x^{2} y - y^{2}} \cdot 6 y x

= y^{2} (1 \cdot e^{3 x^{2} y - y^{2}} + e^{3 x^{2} y - y^{2}} \cdot 6 y xx)

Simplify

y^{2} (1 \cdot e^{3 x^{2} y - y^{2}} + e^{3 x^{2} y - y^{2}} \cdot 6 y xx) : y^{2} (e^{3 x^{2} y - y^{2}} + 6 e^{3 x^{2} y - y^{2}} x^{2} y)

= y^{2} (e^{3 x^{2} y - y^{2}} + 6 e^{3 x^{2} y - y^{2}} x^{2} y)

What is (\partial)/(\partial x)(xy^2e^{3x^2y-y^2}) ?
The answer to (\partial)/(\partial x)(xy^2e^{3x^2y-y^2}) is y^2(e^{3x^2y-y^2}+6e^{3x^2y-y^2}x^2y)