Am I interpreting composite derivatives correctly?

Question

I want to make sure of something:

If I have a function $f(x)$ and I take an arbitrary number of derivatives, $$ \frac{d^{a+b}}{dx^{a+b}}f(x),$$ is that the same as saying

$$\frac{d^{a}}{dx^{a}} \left[ \frac{d^{b}}{dx^{b}}[f(x)] \right]?$$

Answer 1

Yes. If we let $a=1$ and $b=2$, the $(a+b)$th derivative of $f(x)$ is its $3$rd derivative.

You wrote:

$$\frac{d^{a}}{dx^{a}} \left[ \frac{d^{b}}{dx^{b}}[f(x)] \right]$$

Using the values we defined earlier, this would be:

$$\frac{d}{dx} \left[ \frac{d^{2}}{dx^{2}}[f(x)] \right]$$

The derivative of the $2$nd derivative of $f(x)$ is the $3$rd or $(a+b)$th derivative of $f(x)$.