What is (\partial)/(\partial x)(xe^{5xy}) ?

The answer to (\partial)/(\partial x)(xe^{5xy}) is e^{5xy}+5e^{5xy}xy

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(\partial)/(\partial x)(xe^{5xy})

\frac{\partial}{\partial x} (x e^{5 x y})

e^{5 x y} + 5 e^{5 x y} x y

Solution steps

\frac{\partial}{\partial x} (x e^{5 x y})

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = e^{5 x y}

= \frac{\partial}{\partial x} (x) e^{5 x y} + \frac{\partial}{\partial x} (e^{5 x y}) x

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

\frac{\partial}{\partial x} (e^{5 x y}) = e^{5 x y} \cdot 5 y

= 1 \cdot e^{5 x y} + e^{5 x y} \cdot 5 y x

Simplify

= e^{5 x y} + 5 e^{5 x y} x y

\int x^{2} (x^{3} + 9)^{\frac{1}{2}} d x

\int \frac{x}{( x ^{2} + 1 ) ^{3}} d x

\frac{4.5 x - 4}{( 3 x - 4 ) ^{\frac{1}{2}}}

e^{x} + e^{x} x

\frac{d}{d x} (\frac{9}{x ^{3}} + \frac{1}{x})

What is (\partial)/(\partial x)(xe^{5xy}) ?
The answer to (\partial)/(\partial x)(xe^{5xy}) is e^{5xy}+5e^{5xy}xy