What is (\partial)/(\partial x)(xe^{x^3y}) ?

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

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(\partial)/(\partial x)(xe^{x^3y})

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

= 1 \cdot e^{x^{3} y} + e^{x^{3} y} \cdot 3 y x^{2} x

Simplify

1 \cdot e^{x^{3} y} + e^{x^{3} y} \cdot 3 y x^{2} x : e^{x^{3} y} + 3 e^{x^{3} y} x^{3} y

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

\frac{d}{d x} (π x^{2} h)

\int \frac{12}{x ( x ^{2} + 25 )} d x

csc^{2} (x) - 4

(x + 6 x y^{4}) d x + e^{x^{2}} y^{3} d y = 0

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