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

derivative of ln(x^2)

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

The derivative of

\ln(x^2)

\frac{2}{x}

.

Solution. Let

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

f(u) = \ln(u)

and

g(x) = x^2

. We will apply that chain rule:

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

We do know that

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

and that

\frac{d}{dx} x^2 = 2x

. So we get

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

Substituting everything together, we get

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

So the derivative of

\ln(x^2)

\frac{2}{x}