What is (\partial)/(\partial x)(x^yln(x)) ?

The answer to (\partial)/(\partial x)(x^yln(x)) is yx^{y-1}ln(x)+x^{y-1}

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(\partial)/(\partial x)(x^yln(x))

\frac{\partial}{\partial x} (x^{y} ln (x))

y x^{y - 1} ln (x) + x^{y - 1}

Solution steps

\frac{\partial}{\partial x} (x^{y} ln (x))

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{y}, g = ln (x)

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

\frac{\partial}{\partial x} (x^{y}) = y x^{y - 1}

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

= y x^{y - 1} ln (x) + \frac{1}{x} x^{y}

\frac{1}{x} x^{y} = x^{y - 1}

= y x^{y - 1} ln (x) + x^{y - 1}

\frac{\partial}{\partial L} (L^{\frac{1}{2}} K^{\frac{1}{2}})

\frac{\partial}{\partial y} (x + cos (x yz) - 1)

2 x, x^{2 - 3}

\frac{d}{d x} (sin (x) + \frac{2}{9} cot (x))

ln (14 x)

What is (\partial)/(\partial x)(x^yln(x)) ?
The answer to (\partial)/(\partial x)(x^yln(x)) is yx^{y-1}ln(x)+x^{y-1}