**Derivatives**

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

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

*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**

**Exercise 6.3.1**

Typeset the limit definition of the derivative:

**Solution 6.3.1**

**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**

**Higher derivatives**

We typeset higher derivatives using superscripts.

To type the *n*th 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

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