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.
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.
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 ^{t}log(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.
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:
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.