What is (\partial)/(\partial x)(-3xyln(xy)) ?

The answer to (\partial)/(\partial x)(-3xyln(xy)) is -3y(ln(xy)+1)

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(\partial)/(\partial x)(-3xyln(xy))

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

- 3 y (ln (x y) + 1)

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

= - 3 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)

= - 3 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}

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

Simplify

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

= - 3 y (ln (x y) + 1)

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

y^{^{''}} = - y

\int x - 7 d x

x \to + 0 lim (\frac{e ^{x^{2}} - 1}{x})

\frac{d}{d x} (\frac{x ^{2} + 25}{x})

What is (\partial)/(\partial x)(-3xyln(xy)) ?
The answer to (\partial)/(\partial x)(-3xyln(xy)) is -3y(ln(xy)+1)