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Tangents, Derivatives and Differentiation

The Derivative

Let f(x) be a function and assume that for each value of x, we can calculate the slope of the tangent to the graph y = f(x)  at x. This slope depends on the value of x that we choose, and so is itself a function. We call this function the derivative of f(x) and denote it by f ´ (x).

Hence, for a function y = f(x), we denote

Note that f(x+h) − f(x) is the change in y corresponding to the change h in x. Leibniz denoted these changes by Δy and Δx respectively. Thus

For this reason, we often write.

The symbol should be taken as a whole and not treated as “dy divided by dx”. We read it as “dee-y by dee-x”. For clarity, we sometimes write or for the derivative of f(x).

To compute the derivative of a function f(x) is to differentiate the function f with respect to x. The process of finding a derivative is called differentiation.

Example 1.

For a straight line determined by y = mx + c, the tangent of the graph (line) at each point coincides with the line. Thus the tangent has a constant slope m. That is, . We verify this by using the definition of the derivative:

Let f(x) = mx + c. Then

This is independent of h. Hence

In particular, the derivative of a constant function (a line with slope 0) is zero. That is, if c is a constant, then

Example 2.

We find the derivative of f(x) = x².

Hence

More (this will open in another window)

Exercise 1.

Find the derivative of these functions using the definition of the derivative.

<< The Slope of a Tangent Line |  Differentiation Index  |  Derivation Of The Basic Derivative Rules >>