What is the derivative of ln((ax^n)) ?

The derivative of ln((ax^n)) is n/x

What is the first derivative of ln((ax^n)) ?

The first derivative of ln((ax^n)) is n/x

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\frac{d}{d x} (ln ((a x)^{n}))

\frac{n}{x}

Solution steps

\frac{d}{d x} (ln ((a x)^{n}))

Apply the chain rule:

\frac{1}{( a x ) ^{n}} \frac{d}{d x} ((a x)^{n})

= \frac{1}{( a x ) ^{n}} \frac{d}{d x} ((a x)^{n})

\frac{d}{d x} ((a x)^{n}) = n a^{n} x^{n - 1}

= \frac{1}{( a x ) ^{n}} n a^{n} x^{n - 1}

Simplify

\frac{1}{( a x ) ^{n}} n a^{n} x^{n - 1} : \frac{n}{x}

= \frac{n}{x}

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What is the derivative of ln((ax^n)) ?
The derivative of ln((ax^n)) is n/x
What is the first derivative of ln((ax^n)) ?
The first derivative of ln((ax^n)) is n/x