In this chapter we want to summarise the most important examples of derivatives. The derivative rules will allow us for computing derivatives of composite functions.
Now we will calculate some examples of derivatives from the table above. Often it comes down to determining the differential quotient of the function, i.e. a limit value. But sometimes it is also useful to use the calculation rules from the chapter before.
Using the calculation rules for derivatives we can now calculate the derivatives of polynomial functions and rational functions:
Theorem (Derivative of polynomial functions)
Let
with and be a polynomial function of degree . Then is differentiable on all of , and for there is
Proof (Derivative of polynomial functions)
Using the derivative rule for the multiple of a function , every single summand of the polynomial is differentiable on . With the summation rule we can derive every polynomial function term by term on and obtain for :
with the derivative of the zeroth summand disappearing.
In particular, it follows for and that linear and quadratic functions are differentiable onn all of .
Exercise (Derivative of rational functions)
Let
with and a rational function defined on . Show that is differentiable on , and calculate the derivative.
Solution (Derivative of rational functions)
Numerator and denominator of are polynomials. Since the denominator is non-zero on and polynomials are differentiable, it follows from the quotient rule that is differentiable on .
We can already differentiate power functions with natural powers. Now we investigate those with negative integer exponents.
Example (Derivative of the hyperbolic function)
The power function
is differentiable on and there is
for .
Exercise (Derivative of )
Prove that the power function
is differentiable on and compute its derivative.
Solution (Derivative of )
For there is
In the general case with there is
Theorem (Derivative of the power function with negative integer powers)
The power function
is differentiable on , and for there is
Proof (Derivative of the power function with negative integer powers)
For there is
Exercise (Derivative of the power function)
Prove using the quotient rule
Solution (Derivative of the power function)
For there is by the quotient rule
Remark: Of course we can also apply the inverse rule directly, and thus get the same result
Let us look again at the derivatives rule in the last case, i.e. for . If we put , we get . The derivative rule is hence the same as for with . So we can summarize the two cases and get
Theorem (Derivative of the power function with natural powers)
For the power function
is differentiable on . For there is then
In the case of it is even differentiable on all of .
Now we investigate the derivative of root functions. We start again with the simplest case:
Example (Derivative of the square root function)
The square root function
is differentiable on and for there is
Question: Why is the square root function in not differentiable, although it is defined and continuous there?
For the differential quotient there is
So it does not exist. Hence, we have non-differentiability.
Exercise (Derivative of the cubic root function)
Compute the derivative of the cubic root function
Solution (Derivative of the cubic root function)
For there is
Now let us consider the general case of the -th root function. Here there is
Theorem (Derivative of the -th root function)
Let . Then the -th root function
is differentiable on , and for there is
Proof (Derivative of the -th root function)
For there is
This can now be generalised
Theorem (Derivative of the generalized root function)
For and , the generalized root function
is differentiable on , and for there is
Proof (Derivative of the generalized root function)
Since on die functions and are differentiable, the chain rule implies at that
Hint
For and and the power fucniton with rational exponent was defined as
So for we also have the derivative rule
The (generalized) exponential function and generalized power functionsBearbeiten
In this section we prove that the derivative of the exponential function is again the exponential function. So we can determine the derivative of the generalized exponential and power function.
Theorem (Derivative of the exponential function)
The exponential function
is differentiable on , and for there is
How to get to the proof? (Derivative of the exponential function)
For this derivative it is more useful to use the method
Because in this case we know the limit value
Furthermore we need the functional equation of the exponential function
Proof (Derivative of the exponential function)
For there is
Using the chain rule, the derivatives of the generalized exponential function for and the generalized power function for can be calculated:
Theorem (Derivative of the generalized exponential function)
For the generalized exponential function
is differentiable on , and for there is
Proof (Derivative of the generalized exponential function)
For there is
Theorem (Derivative of the generalized exponential function)
For the generalized exponential function
is differentiable on , and for there is
Exercise (Derivative of the generalized exponential function)
Prove that the derivative of the generalized power function at is .
Proof (Derivative of the generalized exponential function)
Now we turn to the derivative of the natural and generalised logarithm function. Since the natural logarithm is the inverse of the exponential function, we can deduce its derivative directly from rule for derivatives of inverse function:
Theorem (Derivative of the natural logarithm function)
The natural logarithm function
is differentiable on . For there is
Proof (Derivative of the natural logarithm function)
For the exponential function there is: . So the function is differentiable, and because of strictly monotonously increasing. Furthermore, is surjective. The inverse function is the (natural) logarithm function
From the theorem about the derivative of the inverse function we now have for every :
The derivative can also be calculated directly using the differential quotient. If you want to try this, we recommend the corresponding exercise (missing).
Using the derivative of the natural logarithm function we can now immediately conclude
Theorem (Derivative of the generalized logarithm function)
For the generalized logarithm function
is differentiable on . For there is
Proof (Derivative of the generalized logarithm function)
From the derivative rule for the multiple of a function, we get that for all :
If the derivative of the natural logarithm is not available, we can calculate it using the theorem of the derivative of the inverse function.
Theorem (Derivative of the arcsin/arccos function)
The inverse functions of the trigonometric functions , are differentiable with
Note: and are defined and continuous on , but only differentiable on .
Proof (Derivative of the arcsin/arccos function)
Derivative of :
For the sine function there is: . So the function is differentiable, and since for all , it is strictly monotonously increasing on this interval. Further, . So is surjective. The inverse function is the arc sine function
From the theorem about the derivative of the inverse we now have for every :
Derivative of :
For the cosine function there is: . So the function is differentiable, and because of , strictly monotonously decreasing. Further, . So is surjective. The inverse function
is differentiable according to the theorem about the derivative of the inverse function, and for every there is:
Theorem (Derivative of the arctan/ arccot function)
The inverse functions of the trigonometric functions , are differentiable, and there is
Proof (Derivative of the arctan/ arccot function)
For the tangent function there is: . So the function is differentiable and strictly monotonically increasing. Further, . So is surjective. The inverse function