What is (\partial)/(\partial x)(xln(x/y)) ?

The answer to (\partial)/(\partial x)(xln(x/y)) is ln(x/y)+1

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(\partial)/(\partial x)(xln(x/y))

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

ln (\frac{x}{y}) + 1

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = ln (\frac{x}{y})

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

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

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

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

Simplify

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

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

\int cos (2 x) sin (x) d x

\int (∣ x ∣ + 1) d x

\frac{x ^{2}}{1 + x}, 0

8 (sin (t) y^{^{'}} + cos (t) y) = cos (t) sin^{5} (t)

\frac{1}{1 + 5 x ^{2}}

What is (\partial)/(\partial x)(xln(x/y)) ?
The answer to (\partial)/(\partial x)(xln(x/y)) is ln(x/y)+1