What is (\partial)/(\partial x)(xln(x+2y)) ?

The answer to (\partial)/(\partial x)(xln(x+2y)) is ln(x+2y)+x/(x+2y)

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(\partial)/(\partial x)(xln(x+2y))

\frac{\partial}{\partial x} (x ln (x + 2 y))

ln (x + 2 y) + \frac{x}{x + 2 y}

Solution steps

\frac{\partial}{\partial x} (x ln (x + 2 y))

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = ln (x + 2 y)

= \frac{\partial}{\partial x} (x) ln (x + 2 y) + \frac{\partial}{\partial x} (ln (x + 2 y)) x

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

\frac{\partial}{\partial x} (ln (x + 2 y)) = \frac{1}{x + 2 y}

= 1 \cdot ln (x + 2 y) + \frac{1}{x + 2 y} x

Simplify

1 \cdot ln (x + 2 y) + \frac{1}{x + 2 y} x : ln (x + 2 y) + \frac{x}{x + 2 y}

= ln (x + 2 y) + \frac{x}{x + 2 y}

\frac{1}{s ^{2} + 6 s + 34}

1 - e^{- t}

cos (x), sin (x), [0, \frac{π}{4}]

\int \frac{7}{( x ^{2} + 1 ) ^{\frac{3}{2}}} d x

\int_{0}^{4} \frac{x}{x ^{2} - 9} d x

What is (\partial)/(\partial x)(xln(x+2y)) ?
The answer to (\partial)/(\partial x)(xln(x+2y)) is ln(x+2y)+x/(x+2y)