What is the Derivative of ln^2(x)?

derivative of ln^2(x)

What is the derivative of ln^2(x)?

The derivative of

\ln^2(x)

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

. To see why we will apply the chain rule.

Solution. Let

h(x) = \ln^2(x)

f(u) = u^2

and

g(x) = \ln(x)

. The chain rule will be the most straightforward property to use:

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

We have seen here that

\frac{d}{dx} \ln(x) = \frac{1}{x}

. So we get

\begin{align*} f'(u) = 2u \quad \text{and} \quad g'(x) = \frac{1}{x}. \end{align*}

Wrapping everything together, we will get the next equality:

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

So, we have that

h'(x) = \frac{2\ln(x)}{x}