Although Isaac Newton and Gottfried Leibniz made major contributions to the concept of the derivative, its development stretches back to the ancient Greeks.

The origin of the derivative goes back to the classical problem of finding the tangent line to a curve.

A line is tangent to a curve at some point P if the line has the same direction as the curve at that point.

Around 300 BC, Euclid found that a tangent line to a circle intersects the circle at only one point. But is this the case for a line tangent to any curve?

Here we have a curve with a tangent line drawn to the it. Use the slider to vary the value of x, and thus move the tangent line along the curve. Observe the tangent line. Does the tangent line always intersect the curve at only a single point?

We see that although there is only a single tangent point P, the line can intersect the curve at a second, non-tangent point Q.

One of the reasons why Newton and others developed calculus was to determine the slope of a line tangent to a curve, illustrated here by the line through point P.

Now, let's add another line which passes through the points P and Q on the curve. This line is known as a secant line.

As we move point Q closer to point P, the slope of the secant line approaches that of the tangent line.

We can calculate the slope of the secant line as shown. We see that as Dx becomes smaller, the slope of the secant line provides a better estimate of the slope of the tangent line.

Therefore, in the limit as Dx → 0, the slope of the tangent line is given by the slope of the secant line.

A limit of this form is found in many real life situations. We call it the derivative.

In general, the derivative of a function f at a point (a,f(a)), denoted by f'(a), is given by the limit shown as the interval h → 0.

What is the derivative of this function at x = a? Click the "Submit" button after entering your response.

Sometimes, it is not possible to determine an exact value for the limit at a point. This is true for the given function when a = 0.

After substituting the function in the limit equation, we see that the denominator would go to zero in the limit as h approaches zero.

In such a case, we can estimate the derivative numerically. We do this by calculating approximate values of the limit for decreasingly smaller intervals.

We make a table of the values, approaching the point from both sides. We see from the table, that the values seem to approach a particular value, which becomes our estimated value.

Now, let's try an example. Estimate the derivative of this function to three decimal places. Start by determining the values in the second column of the table. Click the "Submit" button after entering your estimate.

There is another way we can estimate the derivative at a point P. Click the button to zoom in on the curve at point P. Notice the apparent shape of the curve as you zoom in the maximum amount. Can you guess how we can estimate the derivative?

Notice that as we zoom in, the curve looks more like a straight line.

We can estimate the derivative at point P, by estimating the slope of this apparent 'straight line' from the graph.

What is the derivative of the function estimated from the slope of the line at point P?

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education