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Derivatives of Elementary Functions

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Derivatives of Elementary Functions

Derivatives of Elementary Functions

Derivatives of Elementary Functions

Derivatives of Elementary Functions

Derivatives of Elementary Functions

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.