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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

\int \frac{cos ( 2 x )}{x} d x

\int 8 cos (θ) cos^{5} (sin (θ)) d θ

\frac{d}{d θ} (16 sin (θ))

\frac{d}{d x} (\frac{1}{5 x + 2})

(ln (3 x))^{2}

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