Derivative
The derivative of a function f (x) is another function denoted or f '(x) that measures the relative change of f (x) with respect to an infinitesimal change in x. If we start at x = a and move x a little bit to the right or left, the change in inputs is ∆x = x  a, which causes a change in outputs ∆x = f (x)  f (a). When the change in inputs ∆x = x  a, which causes a change in outputs ∆f = f (x)  f (a). When the change in inputs ∆x = x  a is small enough, the derivative of f at point a is approximated by the ratio:
Notice that this ratio measures how large ∆f = f (x)  f (a), the difference in output, is, compared to ∆x = x  a, the difference in input. In the diagram below, as x approaches a, the ratio approaches the true value of the derivative of f at a.
The derivative of f at a, denoted or f '(a), is defined as
If this limit exists, f is said to be differentiable at a. If f is differentiable at every point in an open interval I, then we say that f is differentiable on I. Geometrically, is the slope of the line tangent to the graph of x = a. We can think of the following diagram as what happens when we move the point (x, f(x)) from the previous diagram closer and closer to (a, f(a)).
At an arbitrary point x, the derivative measures the relative change in f(x) as x moves by a small amount, h, right or left. This time, the change in x is ∆x = h which causes the change ∆f = f (x + h)  f (x) in f. Therefore, an alternate definition of the derivative at x is:
Note: Since the derivative measures the relative change, or ratio of changes, between f and x, it makes sense why we use the notation because of df (x) stands for a small difference, or change, in f (x) and dx stands for a small difference or change in x.
Example: Derivative of f(x) = x^{2}:
If f (x) = x^{2}, then using the h → 0 definition of the derivative,
=  
=  
= 
Now that every term in the numerator and denominator has a factor of h, we can cancel the h's out to get:
Therefore,
This means that the slope of the tangent line of f (x) = x^{2} at say, x = 1, would be :
As a generalization of f (x) = x^{2}, the derivative of f (x) = x^{n}, where n is any real number, is:
The above is called the power rule.
Example: Derivative of f(x) = sin(x)
If f (x) = sin(x), then we use the h → 0 definition of the derivative to get:
= 
We can then use the sum of angles trignonometric identity to expand sin(x + h) into:
Using this identity and continuing from above,
= 
Notice that cos(h) = cos(0) = 1, so we can replace the sin(x)cos(h) above with sin(x) to get:
=  
= 
Given the special limit:
,
we can determine the limit as
which implies:
Table of derivative rules
Function  Derivative 

(power rule) 

for some constant a > 0 and a ≠ 1 

for some constant a > 0 

for any constant c and function f 

for any functions f and g 

product rule 

quotient rule 

the denotes composition, not multiplication 
chain rule 
Undefined derivatives
Note: From here on, whenever we say "the slope of the graph of f at x," we mean "the slope of the line tangent to the graph of f at x."
In some cases, the derivative of a function f may fail to exist at certain points on the domain of f, or even not at all. That means at certain points, the slope of the graph of f is not welldefined. Some cases where the slope is not welldefined include:
1. Discontinuous functions
The function above has no slope at a because the slope from the right does not match the slope from the left.
2. Functions with a sharp turn
The function f(x) = x has a sharp turn at x = 0, where the slope suddenly switches from 1 on the left to 1 on the right.
3. Functions with a vertical tangent or infinite slope
The slope of approaches ∞ from both sides of x = 0.
Did you know??
The Weierstrass function is continuous everywhere but differentiable nowhere! The Weierstrass function is "infinitely bumpy," meaning that no matter how close you zoom in at any point, you will always see bumps. Therefore, you will never see a straight line with a welldefined slope no matter how much you zoom in.