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Derivatives of Elementary Functions

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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 .

Derivatives of Linear Functions

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 .

Derivatives of Polynomial Functions

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.