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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\int \frac{- 4}{3 x ^{2}} d x

x \to 3 lim (\frac{x - 1}{3 - x})

x^{y} + 1

f (x) = e^{2 x} - x^{2} - 1

ln (x^{7})

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