What is (\partial)/(\partial y)(y(1+ln(xy))) ?

The answer to (\partial)/(\partial y)(y(1+ln(xy))) is ln(yx)+2

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(\partial)/(\partial y)(y(1+ln(xy)))

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

ln (y x) + 2

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

= ln (y x) + 2

t y^{^{'}} + t y = 1 - y, y (1) = 0

\int \frac{1}{x - 1} d x

(x^{3} - 5 x) (4 x^{2} - 3 x)

x + k \cdot x

\frac{d}{d x} (\frac{2 - 3 x}{7 - x})

What is (\partial)/(\partial y)(y(1+ln(xy))) ?
The answer to (\partial)/(\partial y)(y(1+ln(xy))) is ln(yx)+2