What is (\partial)/(\partial x)(xln(x+2y+3z)) ?

The answer to (\partial)/(\partial x)(xln(x+2y+3z)) is ln(x+2y+3z)+x/(x+2y+3z)

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(\partial)/(\partial x)(xln(x+2y+3z))

\frac{\partial}{\partial x} (x ln (x + 2 y + 3 z))

ln (x + 2 y + 3 z) + \frac{x}{x + 2 y + 3 z}

Solution steps

\frac{\partial}{\partial x} (x ln (x + 2 y + 3 z))

Treat

y, z

as constants

Apply the Product Rule:

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

f = x, g = ln (x + 2 y + 3 z)

= \frac{\partial}{\partial x} (x) ln (x + 2 y + 3 z) + \frac{\partial}{\partial x} (ln (x + 2 y + 3 z)) x

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

\frac{\partial}{\partial x} (ln (x + 2 y + 3 z)) = \frac{1}{x + 2 y + 3 z}

= 1 \cdot ln (x + 2 y + 3 z) + \frac{1}{x + 2 y + 3 z} x

Simplify

1 \cdot ln (x + 2 y + 3 z) + \frac{1}{x + 2 y + 3 z} x : ln (x + 2 y + 3 z) + \frac{x}{x + 2 y + 3 z}

= ln (x + 2 y + 3 z) + \frac{x}{x + 2 y + 3 z}

\frac{d}{d x} (\frac{csc ( 3 x )}{x ^{3} + 1})

\int t e^{2 t} d t

\int sin^{4} (3 t) d t

\frac{\partial}{\partial x} (- \frac{2}{1 - 2 x})

\frac{\partial}{\partial x} (x^{2} (y - x))

What is (\partial)/(\partial x)(xln(x+2y+3z)) ?
The answer to (\partial)/(\partial x)(xln(x+2y+3z)) is ln(x+2y+3z)+x/(x+2y+3z)