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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

f (x) = 4 x ln (x) - 7 x

\int \frac{1}{sinh ( x )} d x

\int 2 x^{2} - 5 d x

(x^{2} \cdot 6^{x})^{3}

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