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

derivative of log^2(x) base a

What is the derivative of log^2(x) base a?

To be specific in this proof, we will have a logarithm

x

with base

a

, i.e.,

\log_a^2(x)

. We will see that the derivative of

\log_a^2(x)

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

.

Proof. Let

F(x) = \log_a^2(x)

f(u) = u^2

and

g(x) = \log_a(x)

such that

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

. We will use the chain rule:

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

We can see here that

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

. So we have

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

Substituting all the results, we get

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

So we have that the derivative of

\log_a^2(x)

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