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Another key aspect to understanding calculus the derivative. After one has learned to take a limit, taking a derivative can be approached. A point on a graph where there is a limit are called "points of differentiability"(11). If the point x on function f is differentiable , then it is said that the derivative of the function is at x The derivative is the primary tool that is used to calculate rates of change and slopes of tangent lines in graphs. The derivative can be defined f' of function f can be defined as a function whose slope at x is the slope of the tangent line at the point of the graph of f(x), or a derivative can be interpreted as the instantaneous rate of change of the function f at point x. The key concept behind a derivative is to take the difference of a point in time x and after a short while, h, where one will be, f(x+h): (f(x + h) - f(x))/h. If you take the limit of the function as h approaches zero, then you have derived the function at the value x, and the new function is the derivative.

The above illustration of a graph may help you better discern what a derivative is. The variable x is any number along the X axis. The variable h is a distance (the distance from x). It follows that the distance from x to (x + h) is h. That is called the run of the triangle. You get the rise by figuring out the function at x, then evaluating it again, this time at (x + h), and subtracting one from the other. So, the distance between f(x) and f(x + h) is the rise of our triangle. The slope of the segment that we used to form our triangle is the rise divided by the run. The picture shows us that one is finding the length of a segment ( the green brace). It is easy to see that if we made smaller triangles, we would get a more precise measurement of the slope. This is why we make h--->0.

One of the most basic examples of a derivative being used in everyday life involves the equation that models the change of a population as being proportional to the number of individuals in the population. In symbols, if P(t) represents the number of individuals in a population at time t, then the so-called exponential growth model is: dP/dt = k P.
Recall that the general solution of this differential equation is of the form C exp(kt) where C is the population at the time that we first consider it, and where "exp" is just another way to write "ee-to-the." Recall also that in order to get a particular solution, we must have some sort of experimental observations that tell us
1. the initial population, and
2. the net birth rate.
The sign of k determines whether the population will grow without limit, or whether it will become extinct(13).

Another example of how a derivative can be used in the real world is to figure out, say, when the part of an airplane is under the most stress. A function that shows the stresses on a particular airplane part with respect to time in flight is: f(t) = -t2 + 2 t + 3
If you take the derivative of the function, you get:
f'(t) = -2t + 2.
Then determine that the derivative was equal to zero at t = 1(12).
So the time of most stress was 1 minute into the flight. Another use for the derivative of a function is finding the maximum or minimum points of the function. The derivative indicates the rate of change of a function, so the derivative measures the slope:

"When the slope changes from positive to negative, the function is at its maximum when the slope is zero. When the slope changes from negative to positive, it is at its minimum when the slope is zero"(12).

The derivative is a slope in its most simple terms. If you have a function f(x), then its derivative (f'(x)) would be the slope of the line that is tangent to some curve of the graph of f(x). If the graph is a straight line, or linear function, then the derivative would simply be the slope of the graph, which leads to the realization that "derivative of a straight line is everywhere equal to its slope"(12). A derivative is also a rate. It is simply a rate that can change constantly with time or with some other variable. Because the world is full of variables that have constantly changing rates, understanding derivatives will allow you to understand the world more clearly. The derivative allows one to find the slope of a tangent line of the graph, but what about finding the area of a graph?

What are derivatives?