What is (\partial)/(\partial x)(xln(1+x+y)) ?

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

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(\partial)/(\partial x)(xln(1+x+y))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = ln (1 + x + y)

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

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

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

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

Simplify

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

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

\int_{0}^{1} x d x

e^{x} (7 x^{2} - 14 x + 14)

\frac{4}{x ^{2}} - \frac{1}{x}

\frac{d ^{2}}{d x ^{2}} (\frac{1}{1 - x ^{2}})

\int 2 y arctan (x) d y

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