What is (\partial)/(\partial x)(axln(ax)) ?

The answer to (\partial)/(\partial x)(axln(ax)) is a(ln(ax)+1)

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(\partial)/(\partial x)(axln(ax))

\frac{\partial}{\partial x} (a x ln (a x))

a (ln (a x) + 1)

Solution steps

\frac{\partial}{\partial x} (a x ln (a x))

Take the constant out:

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

= a \frac{\partial}{\partial x} (x ln (a x))

Apply the Product Rule:

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

f = x, g = ln (a x)

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

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

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

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

Simplify

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

= a (ln (a x) + 1)

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What is (\partial)/(\partial x)(axln(ax)) ?
The answer to (\partial)/(\partial x)(axln(ax)) is a(ln(ax)+1)