Question

How do you find the derivative of `ln(x^2)`?

Answer 1

`\frac{2}{x}`

Explanation:

Using the chain rule and letting `u(x) = x^2` we have `\frac{d}{dx} \ln ( u(x)) = \frac{d}{du} \ln ( u(x)) . \frac{du}{dx} = \frac{1}{u(x) }2x`

`= \frac{2x}{x^2} = \frac{2}{x}`

Answer 2

`2/x`

Explanation:

There are two methods:

Using the Chain Rule:

Since the derivative of `ln(x)` is `1/x`, we see that the derivative of a function inside the natural logarithm, such as `ln(f(x))`, is `1/f(x)*f'(x)`.

So, for `ln(x^2)`, the derivative is `1/x^2*2x`, since `2x` is the derivative of `x^2`.

Then, we see that `1/x^2*2x` simplifies to `2/x`.

Simplifying first:

Using the rule `log(a^b)=b*log(a)`, we see that `ln(x^2)=2*ln(x)`.

The derivative of `2ln(x)` is just `2` times the derivative of `ln(x)`, which is `1/x`.

We see that `2*1/x=2/x`, the answer we obtained earlier.