What is (\partial)/(\partial x)(x^{10}e^{xy}) ?

The answer to (\partial)/(\partial x)(x^{10}e^{xy}) is 10x^9e^{xy}+e^{xy}yx^{10}

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(\partial)/(\partial x)(x^{10}e^{xy})

\frac{\partial}{\partial x} (x^{10} e^{x y})

10 x^{9} e^{x y} + e^{x y} y x^{10}

Solution steps

\frac{\partial}{\partial x} (x^{10} e^{x y})

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{10}, g = e^{x y}

= \frac{\partial}{\partial x} (x^{10}) e^{x y} + \frac{\partial}{\partial x} (e^{x y}) x^{10}

\frac{\partial}{\partial x} (x^{10}) = 10 x^{9}

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

= 10 x^{9} e^{x y} + e^{x y} y x^{10}

x \to 2 - lim (\frac{2 x + 8}{x + 2})

\frac{d}{d x} (\frac{π x ^{2}}{4})

\int \frac{1}{4 y ^{2} + 12 y + 9} d y

f (t) = \frac{t ^{3}}{3}

\frac{d}{d y} (\frac{d y}{y})

What is (\partial)/(\partial x)(x^{10}e^{xy}) ?
The answer to (\partial)/(\partial x)(x^{10}e^{xy}) is 10x^9e^{xy}+e^{xy}yx^{10}