The concept of **Derivative** is at the core of Calculus and
modern mathematics. The definition of the derivative can be
approached in two different ways. One is geometrical (as a slope
of a curve) and the other one is physical (as a rate of change).
Historically there was (and maybe still is) a fight between
mathematicians which of the two illustrates the concept of the
derivative best and which one is more useful. We will not dwell
on this and will introduce both concepts. Our emphasis will be
on the use of the derivative as a tool.

If we now assume that A and B are very close to each other, we get close to what is called the

The instantaneous velocity (at A) will be found when get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have

If

**Example.** Consider a parabolic motion given by the function
*f*(*t*) = *t*^{2}. The instantaneous velocity at *t*=*a* is given by

Since

we conclude that the instantaneous velocity at

This concept of velocity may be extended to find the rate of change of any variable with respect to any other variable. For example, the volume of a gas depends on the temperature of the gas. So in this case, the variables are V (for volume) as a function of T (the temperature). In general, if we have *y* = *f*(*x*), then the average rate of change of *y* with respect to *x* from *x* = *a* to
,
where
,
is

As before, the instantaneous rate of change of

**Notation.** Now we get to the hardest part. Since we can not
keep on writing "Instantaneous Velocity" while doing computations,
we need to come up with a suitable notation for it. If we write
*dx* for
small, then we can use the notation

This is the notation introduced by Leibniz. (Wilhelm Gottfried Leibniz (1646-1716) and Isaac Newton (1642-1727) are considered the inventors of Calculus.)

Fix a point on the graph, say
(*x*_{0}, *f*(*x*_{0})). If the graph as a
geometric figure is "nice" (i.e. smooth) around this point, it is
natural to ask whether one can find the equation of the straight
line "touching" the graph at that point. Such a straight line is
called the **tangent line** at the point in question. The
concept of tangent may be viewed in a more general framework.

(Note that the tangent line may not exist. We will discuss this
case later on.) One way to find the tangent line is to consider
points (*x*,*f*(*x*)) on the graph, where *x* is very close to
*x*_{0}. Then draw the straight-line joining both points (see the
picture below):

As you can see, when *x* get closer and closer to *x*_{0}, the
lines get closer and closer to the tangent line. Since all these
lines pass through the point
(*x*_{0},*f*(*x*_{0})), their equations will
be determined by finding their slope: The slope of the line
passing through the points
(*x*_{0},*f*(*x*_{0})) and (*x*,*f*(*x*)) (where
)
is given by

The tangent itself will have a slope

In other words, we have

So the equation of the tangent line is

**Notation.** Writing "m" for the slope of the tangent line
does not carry enough information; we want to keep track of the
function *f*(*x*) and the point *x*_{0} in our notation. The common
notation used is

In this case, the equation of the tangent line becomes

where

One last remark: Sometimes it is more convenient to compute limits when the variable approaches 0. One way to do that is to make a translation along the x-axis. Indeed, if we set

**
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