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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:

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.

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