What is (\partial)/(\partial x)(4e^{x^2y}xy) ?

The answer to (\partial)/(\partial x)(4e^{x^2y}xy) is 4y(2e^{x^2y}x^2y+e^{x^2y})

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(\partial)/(\partial x)(4e^{x^2y}xy)

\frac{\partial}{\partial x} (4 e^{x^{2} y} x y)

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

Solution steps

\frac{\partial}{\partial x} (4 e^{x^{2} y} x y)

Treat

y

as a constant

Take the constant out:

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

= 4 y \frac{\partial}{\partial x} (e^{x^{2} y} x)

Apply the Product Rule:

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

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

= 4 y (\frac{\partial}{\partial x} (e^{x^{2} y}) x + \frac{\partial}{\partial x} (x) e^{x^{2} y})

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

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

= 4 y (e^{x^{2} y} \cdot 2 y xx + 1 \cdot e^{x^{2} y})

Simplify

4 y (e^{x^{2} y} \cdot 2 y xx + 1 \cdot e^{x^{2} y}) : 4 y (2 e^{x^{2} y} x^{2} y + e^{x^{2} y})

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

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What is (\partial)/(\partial x)(4e^{x^2y}xy) ?
The answer to (\partial)/(\partial x)(4e^{x^2y}xy) is 4y(2e^{x^2y}x^2y+e^{x^2y})