Derivatives

Recall that there are two popular notations for the derivative, Leibniz notation and Lagrange notation.

Lagrange notation for the derivative.

f'(x) is the Lagrange notation for the derivative of f with respect to x, read "f prime of x."

Leibniz notation for the derivative.

df/dx is the Leibniz notation for the derivative of f with respect to x, read "dee ef dee ex."

To get either notation, we can reuse some familiar commands.

Leibniz notation is a fraction: \frac{df}{dx}

Lagrange notation adds an apostrophe: f'(x)

Exercise 6.3.1

Typeset the limit definition of the derivative:

d f d x equals f prime of x equals the limit as h goes to 0 of f of (x + h) minus f of (x), divided by h

Solution 6.3.1

The limit definition of the derivative can be typeset as follows:

\[ \frac{df}{dx} = f'(x) = \lim_{h\to0}\frac{f(x + h) - f(x)}{h} \]

Higher derivatives

We typeset higher derivatives using superscripts.

To type the nth derivative of f with respect to x...

...in Leibniz notation, add the superscript n to the d in the numerator and to the x in the denominator

\[ \frac{d^{n}f}{dx^{n}} \]

...in Lagrange notation, add (n) as a superscript to f

\[ f^{(n)}(x) \]