What is the Derivative of ln^3(x)?

Derivative of ln^3(x)

The derivative of

\ln^3(x)

\frac{3\ln^2(x)}{x}

.

Solution. Let

F(x) = \ln^3(x)

f(u) = u^3

and

g(x) = \ln(x)

such that

F(x) = f(g(x))

. Using the chain rule, we can determine the derivative of

\ln^3(x)

\begin{align*} F'(x) = f'(g(x))g'(x). \end{align*}

We saw here that

g'(x) = \frac{1}{x}

, and

f'(u) = 3u^2

. So we get:

\begin{align*} f'(g(x)) = 3g(x)^2 = 3\ln^2(x). \end{align*}

Together, we get:

\begin{align*} F'(x) &= f'(g(x))g'(x) \\ &= 3\ln^2(x) \frac{1}{x} \\ &= \frac{3\ln^2(x)}{x}. \end{align*}

So, the derivative of

\ln^3(x)

\frac{3\ln^2(x)}{x}