What is (\partial)/(\partial x)((x+y)ln(x/y)) ?

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

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(\partial)/(\partial x)((x+y)ln(x/y))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\int_{0}^{1} \frac{47}{4 y - 1} d y

y^{^{'}} = 3 - 6 x + y - 2 x y

\frac{1}{( 1 + x ^{2} )}

\frac{d}{d x} (sin (x^{7}))

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