In this section we compute the derivatives of the elementary functions. We use the definition of the derivative as a limit of difference quotients. Recall that a function f is said to be differentiable at a value x in its domain if the limit
exists, and that the value of this limit is called the derivative of f at x .
A linear function has the form f (x) = ax + b . Since the slope of this line is a , we would expect the derivative f'(x) to equal a at every point in its domain. Computing the limit of the difference quotient, we see that this is the case:
f'(x) | = | ||
= | |||
= | |||
= | a | ||
= | a |
Thus the graph of the derivative is the horizontal line f'(x) = a .
Note, as a special case, that the derivative of any constant function f (x) = b is a constant function equal to 0 at every value in its domain: f'(x) = 0 .
We will show in the next section that the derivative of a sum of two functions is equal to the sum of the derivatives of the two functions. For example, considering the linear function f above, let f _{0}(x) = b and f _{1}(x) = ax . Then f (x) = f _{0}(x) + f _{1}(x) , so f'(x) = f _{0}'(x) + f _{1}'(x) = a + 0 = a , agreeing with our previous result.