3.1 - Alternate Definition of the Derivative

Old definitions

We already know one definition of the derivative:

Basic Derivatives

Definition of the derivative

The derivative of the function $f$ with respect to the variable $x$ is the function $f^{'}$ whose value at $x$ is $f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

And we can find derivatives at a point one way:

Basic Derivatives

Slope of a curve at a point

The slope of the curve $y = f (x)$ at the point $(a, f (a))$ is $m = lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$
- (This is also the equation for the slope of the tangent to $y = f (x)$ at that point)
The normal line to a curve at a point is the line perpendicular to a tangent at that point.

Alternate definition of the derivative

The derivative of $f$ at the point $x = a$ is the limit $lim_{x \to a} \frac{f (x) - f (a)}{x - a}$ (provided the limit exists).

For example, you could use this formula to find the derivative of $f (x) = \sqrt{x}$ to be $\frac{1}{2 \sqrt{a}}$ for any $a$ .
- Substituting $x$ in instead of $a$ you could find the general derivative of the function.

Old definitions

Basic Derivatives

Definition of the derivative

Basic Derivatives

Slope of a curve at a point

Alternate definition of the derivative

Graphing Derivatives