We compute the instantaneous growth rate by computing the limit of average growth rates.

Given a function, it is often useful to know the rate at which the function changes. To give you a feeling why this is true, consider the following:
  • If represents the distance (position relative to an origin) of an object with respect to time, the rate of change gives the velocity of the object.
  • If represents the velocity of an object with respect to time, the rate of change gives the acceleration of the object.
  • If represents the revenue generated by selling objects, the rate of change gives us the marginal revenue, meaning the additional revenue generated by selling one additional unit. Note, there is an implicit assumption that is quite large compared to .
  • If represents the cost to produce objects, the rate of change gives us the marginal cost, meaning the additional cost generated by selling one additional unit. Again, there is an implicit assumption that is quite large compared to .
  • If represents the profit gained by selling objects, the rate of change gives us the marginal profit, meaning the additional cost generated by selling one additional unit. Again, there is an implicit assumption that is quite large compared to .
  • The rate of change of a function can help us approximate a complicated function with a simple function.
  • The rate of change of a function can be used to help us solve equations that we would not be able to solve via other methods.

From slopes of secant lines to slopes of tangent lines

You’ve been computing average rates of change in class for a while now, the computation is simply or when finding the average rate of change of a function on the interval .

However, the question remains: Given a function that represents an amount, how exactly does one find the function that will give the instantaneous rate of change? Recall that the instantaneous rate of change of a straight line at any point is the slope of the tangent line. Hence the instantaneous rate of change of a function at a point is the slope of the tangent line of the function at that point. For now, consider the following informal definition of a tangent line:

Given a function and a number in the domain of , if one can ‘‘zoom in’’ on the graph at sufficiently so that it appears to be a straight line, then that line is the tangent line to at the point .

We illustrate this informal definition with the following diagram:

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The derivative of a function at , is the instantaneous rate of change, and hence is the slope of the tangent line of at .

What is the instantaneous rate of change of at ?
The rate of change of a straight line at any point is the slope of the tangent line.
The line tangent to at any point is, simply, itself ()!

The derivative is .

Unfortunately, if is not a straight line, we cannot use the slope to calculate this rate of change of at a point, since is the only point on this line that we know. In order to deal with this problem, we consider secant lines, lines that locally intersect the curve at two points. One of these points will be , the point at which we are trying to find the rate of change. If we call the difference between the -coordinates of the two points, then the second point for our secant line is . The slope of any secant line that passes through the points and is given by

The following diagram shows the secant lines for several values of , as well as the tangent line at .

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Notice that as approaches i.e. approaches , the slopes of the secant lines are approaching the slope of the tangent line. This leads to the definition of the derivative:

Which of the following computes the derivative, ?

Now we will give a number of examples.