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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).


The derivative of f(x) at the point x is equal to the slope of the tangent
to y = f(x) at x

(if the slope exists and is finite).


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


f'(x)= the limit as h tends to zero of ((f(x+h)-f(x))/h

and call it the derivative of f(x) if the limit exists and is finite.



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

f'(x)= the limit as delta x tends to zero of delta y divided by delta x.

For this reason, we often write.


The symbol dy/dx 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 df(x)/dx or (d/dx)f(x) 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, dy/dx=m . We verify this by using the definition of the derivative:

Let f(x) = mx + c. Then


This is independent of h. Hence

dy/dx= the limit as h tends to zero of (f(x+h)-f(x))/h=m=f'(x).

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.




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 >>


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