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Computing Derivatives

Derivatives of Elementary Functions

Terms

Problems

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.

In studying polynomial functions, it is therefore enough to find the derivative of a monomial function of the form f (x) = ax n . Plugging into the formula for the derivative, we have


f'(x) =  
  =  
  =  
  = a[nx n-1 + x n-2 Δx + ... + Δx n-1]  
  = anx n-1  

Thus, to take the derivative of a monomial function, we multiply by the exponent and reduce the exponent by 1 . Using the property of the derivative mentioned above, we see that the derivative of the polynomial function f (x) = a n x n + ... + a 1 x + a 0 is given by f (x) = na n x n-1 + ... + a 2 x + a 1 .

We will wait until we have the quotient rule at our disposal before we calculate the derivatives of rational functions.

Derivatives of Power Functions

A power function has the form f (t) = Cr t . Plugging into the formula for the derivative, we have


f'(t) =  
  =  
  =  
  = Cr t  

The limit in the final expression above does not depend on t , so it is a constant. In fact, this limit is one way of defining the value of the natural logarithm function at r , or log(r) . Thus we have

f'(t) = Cr tlog(r)    

In the special case where r = e , where e is the number such that log(e) = 1 , we have f'(t)=f(t). The functions f (t) = Ce t are the only functions that are equal to their own derivatives.

Derivatives of Trigonometric Functions

We now give one way of calculating the derivative of the sine function. Let f (x) = sin(x) . Using the trigonometric identity sin(a + b) = sin(a)cos(b) + sin(b)cos(a) , we have


f'(x) =  
  =  
  =  
  = sin(x) + cos(x)  
  = cos(x)  

where the last equality follows from examining the figure below:

Figure %: Calculating the Derivative of the Sine Function

We may similarly compute the derivative of g(x) = cos(x) to be g'(x) = - sin(x) . Finally, since tan(x) = sin(x)/cos(x) , it will follow from the quotient rule that the derivative of h(x) = tan(x) is h'(x) = 1/(cos(x))2 .

We will compute the derivatives of the inverse trigonometric functions in the next section, using implicit differentiation.

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