What is (\partial)/(\partial x)(x^5e^{3xy}) ?

The answer to (\partial)/(\partial x)(x^5e^{3xy}) is 5x^4e^{3xy}+e^{3xy}*3yx^5

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(\partial)/(\partial x)(x^5e^{3xy})

\frac{\partial}{\partial x} (x^{5} e^{3 x y})

5 x^{4} e^{3 x y} + e^{3 x y} \cdot 3 y x^{5}

Solution steps

\frac{\partial}{\partial x} (x^{5} e^{3 x y})

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{5}, g = e^{3 x y}

= \frac{\partial}{\partial x} (x^{5}) e^{3 x y} + \frac{\partial}{\partial x} (e^{3 x y}) x^{5}

\frac{\partial}{\partial x} (x^{5}) = 5 x^{4}

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

= 5 x^{4} e^{3 x y} + e^{3 x y} \cdot 3 y x^{5}

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

\int \frac{1}{4 + ( x - 2 ) ^{2}} d x

\frac{d y}{d t} = k y^{2}, y (0) = 47

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

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

What is (\partial)/(\partial x)(x^5e^{3xy}) ?
The answer to (\partial)/(\partial x)(x^5e^{3xy}) is 5x^4e^{3xy}+e^{3xy}*3yx^5