## Derivatives

The deriviative of f(x) at x=a is given by

The quotient represents the slope of the line (called the secant line)
through the points (a, f(a)) and (a+h, f(a+h)):

Consider the function f(x) = x^{2} at x = 0.5:

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Investigate the Limit as h Approaches 0 from the Left

Let's calculate the slope of the secant line for h = -1.5, -1.0, and -0.5:

These secant lines (and their slopes) are shown on the following graph:

As h gets smaller and smaller (approaching 0 from the left), the secant line
pivots around and gets closer and closer to the tangent line shown in red.
That is,

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Investigate the Limit as h Approaches 0 from the Right

Let's calculate the slope of the secant line for h = 1.5, 1.0, and 0.5:

These secant lines (and their slopes) are shown on the following graph:

As h gets smaller and smaller (approaching 0 from the right), the secant
line gets closer and closer to the tangent line shown in red. That is,

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Conclusion

Since both one-sided limits exist and are equal we can assert that

Therefore, the derivative of f(x) = x^{2} at x = 0.5 is 1. That is,
f '(0.5) = 1. The animated graphic below illustrates the secant lines as
h approaches zero from the left and from the right. In both cases, the
secant line approaches the tangent line which has a slope of one.

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A Generalization

Rather than finding the limit of the slope at some particular value of x,
let's find the limit in terms of x:

The slope of the line tangent to the curve f(x) = x^{2} at any value of x is given by f''(x) = 2x.
In the graph below, several lines tangent to the curve f(x) = x^{2} are shown (with the corresponding slopes).
Notice that, in every case, the slope = 2x where x is the x-coordinate of the corresponding point on the curve.