What is (\partial)/(\partial x)(x*ln(x-y)) ?

The answer to (\partial)/(\partial x)(x*ln(x-y)) is ln(x-y)+x/(x-y)

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(\partial)/(\partial x)(x*ln(x-y))

\frac{\partial}{\partial x} (x \cdot ln (x - y))

ln (x - y) + \frac{x}{x - y}

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = ln (x - y)

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

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

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

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

Simplify

1 \cdot ln (x - y) + \frac{1}{x - y} x : ln (x - y) + \frac{x}{x - y}

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

sin (t) - t cos (t), at t = π

\int_{b}^{2 b} x^{9} d x

\int e^{14 x} d x

\frac{d}{d x} ((x^{2} + 3) (x^{2} - 4 x))

x \to - 3 + lim (\frac{1}{6 + 2 x})

What is (\partial)/(\partial x)(x*ln(x-y)) ?
The answer to (\partial)/(\partial x)(x*ln(x-y)) is ln(x-y)+x/(x-y)