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 f(x) and we differentiate it. How do we write the derivative?

Because f(x) is a function, we can write this as:

f'(x)

Notice the apostrophe after the f. That indicates the derivative.

Writing the derivative of an expression

If we instead have an expression, like x^2 + 3x + 2, we can write the derivative in a different way:

\frac{dy}{dx}(x^2 + 3x + 2)

\frac{dy}{dx} indicates ‘the derivative of y with respect to x’. Basically, we’re saying ‘how does y change as x changes?’.

Higher-order derivatives

Higher-order derivatives of a function

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

g''(x)

Similarly, the third derivative would be:

g'''(x)

… 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 y with respect to x is written as:

\frac{d^2y}{dx^2}

The third derivative would be:

\frac{d^3y}{dx^3}

… 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 d (e.g. d^2y) in the numerator, and after the x (e.g. dx^2) in the denominator. This is just something you need to remember.

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