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

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

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(\partial)/(\partial x)(x*ln(x+2y))

\frac{\partial}{\partial x} (x \cdot 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}

y^{^{'}} = \frac{4 t + y}{4 y - t}

\frac{\partial}{\partial y} (sin (x^{2} + y^{2}))

\frac{d y}{d x} = \frac{2 x ( y - 1 )}{x ^{2} + 3}

\int \frac{1 + cos ( x )}{1 + sin ( x )} d x

\frac{\partial}{\partial x} (5 x^{4} + 2 x y + 1)

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