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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:
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 f0(x) = b and f1(x) = ax. Then f (x) = f0(x) + f1(x), so f'(x) = f0'(x) + f1'(x) = a + 0 = a, agreeing with our previous result.
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