We begin with the familiar notion of the tangent to a circle, depicted below:

Calculus, to a certain extent, concerns itself with the study of tangents to a curve. Depicted below is the graph of a polynomial function with tangents drawn at various points:

Upon observation, two important properties of the tangents to a curve may become apparent:

1) At the point at which it is tangent to the curve, the tangent line touches the curve, but does not "cross" it. This is to say that tangent lines are distinct from lines such as the one below, which also touches the graph at only one point, but which clearly "crosses" it:

2) The second important property of a tangent line is that it has the same slope as the point of the graph that it touches. Although a formal definition for the slope of a curve at a point has not yet been presented, it should be visually clear that the slope of the tangent line matches the slope of the curve at the point of tangency.

"Slope" is a concept that can easily be applied to linear functions. It is the change in y divided by the change in x. To calculate the slope of a line, we pick any two points on that line and divide the difference in their y-values by the difference in their x- values.

Any two points can be used in determining the slope of a line, because the slope is constant throughout. Now consider the challenge of trying to find the slope of the following figure:

It should be readily apparent that there is no single slope for this figure. Instead, the curve has a different slope at each separate point. Therefore, for non-linear figures, it makes sense only to talk about the slope at a particular point.

Example: Find the slope of the graph of f at an arbitrary point x.

To visualize what needs to be done, let's consider an arbitrary function f and delineate an arbitrary point x:

The question asks us to find the slope of f at this arbitrary point x. The method with which we are already familiar calls for picking two points on the curve and calculating , so lets proceed this way first. Clearly, one of the points we should use is the point (x, f (x)), since this is the point on the graph where we want to find the slope. But what should be chosen as the other point? Intuitively, it might seem that no other point would yield the right answer, since we are interested in the slope at the single point (x, f (x)) only. Nevertheless, let us pick an arbitrary point h units away on the x-axis, (x + h, f (x + h)):

Now, we can calculate the quantity for these two points:

This quantity,

is called the difference quotient. It does not represent the slope of the graph at (x, f (x)). Rather, it represents the slope of the secant line that goes through the points (x, f (x)) and (x + h, f (x + h)):

To get the slope of the curve at the point (x, f (x)), let's now draw the tangent line at (x, f (x)).

Recall that the tangent to the graph has the same slope as the graph at the point of tangency. Therefore, finding the slope of the graph at (x, f (x)) is the same as finding the slope of the tangent line we have just drawn.

Now comes a crucial step. Consider what happens to the secant line as h, the distance between the two points on the x-axis, is made progressively smaller:

It appears now that as h gets smaller, the secant line looks more and more like the tangent line, which means that the slope of the secant is getting closer and closer to the slope of the tangent. This suggests that if we could make h arbitrarily small, the slope of the secant would get arbitrarily close to the slope of the tangent. Using limits, this idea could be represented as:

Substituting in the difference quotient for the slope of the secant yields

Since the slope of the tangent is the same as the slope of the graph at the point of tangency, we can say:

This is one of the central ideas of all of calculus. The limit of the difference quotient is such an important expression that it is given a name, the derivative, and is represented by "f'(x)". Thus, we can say:

is the derivative of the function f with respect to x.

The derivative gives the slope of the curve (also the slope of the tangent to the curve) at the point (x, f (x)). The derivative itself is also a function, because for every x value that it is given, it returns a value that is equal to the slope of the tangent to f at x.

An alternate notation for the derivative is the Leibniz Notation, when means "the derivative of whatever follows with the respect to x". Thus, means the derivative of f with respect to x, or f'(x) = means the derivative of y with respect to x. Since y commonly means f (x), this is usually the same as

A function f is said to be differentiable at x = a if f'(a) exists. In other words, a function is differentiable at x = a if

exists.

Intuitively, for a function to be differentiable, it needs to be both continuous and "smooth". What is meant by "smooth" is that there are no sharp turns in the graph.

Tangent lines can only be drawn to graphs in places where they are both continuous and smooth, as shown below:

One example of a function that is continuous but not "smooth" throughout is the absolute value function. Consider f (x) =|x|. This function is continuous, but has a sharp "corner" at x = 0:

The function f (x) =|x| is not differentiable at x = 0 because the sharp corner makes it impossible to draw a single tangent line, since there is no defined slope there. Thus, f'(0) does not exist for this function.

Note that any differentiable function must also be continuous, since it is impossible to have a defined slope at a point of discontinuity. However, not all continuous functions are differentiable. An example of this was seen with the absolute value function.