Derivative notation

We have a few options for writing the derivative of a function or expresssion.

Writing the derivative of a function

Let’s say we have a function and we differentiate it. How do we write the derivative?

Because is a function, we can write this as:

Notice the apostrophe after the . That indicates the derivative.

Writing the derivative of an expression

If we instead have an expression, like , we can write the derivative in a different way:

indicates ‘the derivative of with respect to ’. Basically, we’re saying ‘how does change as changes?’.

Higher-order derivatives

Higher-order derivatives of a function

If we want to write, for example, the second derivative of a function , we can use double apostrophes (one for each order of derivative):

Similarly, the third derivative would be:

… and so on.

Higher-order derivatives of an expression

For expressions, it’s a little more that we need to do, but nothing complicated. The second derivative, for example, of an expression for with respect to is written as:

The third derivative would be:

… and again, the pattern repeats (just increase the power for each order of derivative).

Note that, for higher-order derivatives of expressions, we write the power after the (e.g. ) in the numerator, and after the (e.g. ) in the denominator. This is just something you need to remember.