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.
In the following table , and is given. We also define , and .
function term
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term of the derivative function
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domain of definition of the derivative
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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.
We start with some simple derivatives:
Theorem (Derivative of a constant function)
Every constant function is differentiable on all of with derivative .
Proof (Derivative of a constant function)
Let . Then there is
Power functions with natural numbers as powers
[Bearbeiten]
Now we turn to the derivative of power functions with natural powers. First we will deal with a few special cases:
Example (Derivative of the identity and the square function)
The functions
and
are differentiable on all of . Further there is for :
as well as
For the derivative of we used the 3rd binomial formula .
Exercise (Derivative of a power function)
Compute die derivative von
Solution (Derivative of a power function)
For there is
Instead of using the identity , we could also have calculated using polynomial division.
Now we turn to the general case, i.e. the derivative of for :
Proof (Derivative of power functions)
For , so there is
We used the geometric sum formula and the continuity of the polynomial function .
Using the calculation rules for derivatives we can now calculate the derivatives of polynomial functions and rational functions:
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
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
Proof (Derivative of the -th root function)
For there is
This can now be generalised
The (generalized) exponential function and generalized power functions
[Bearbeiten]
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:
Proof (Derivative of the generalized exponential function)
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)
For the chain rule yields
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
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
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 sine function)
The sine function is differentiable. For all there is:
Proof (Derivative of the sine function)
For there is
Theorem (Derivative of the cosine function)
The cosine function is differentiable with
Proof (Derivative of the cosine function)
Theorem (Derivative of the tangent function)
The tangent function
is differentiable on , and for there is
Exercise (Derivative of the cotangent function)
The cotangent function
is differentiable on , and for there is
The derivatives of secant and cosecant can be found in the corresponding exercise.
Using the rule for derivatives of the inverse function we can differentiate the arc-functions (which are inverses of sine, cosine, etc.)
Theorem (Derivative of the arcsin/arccos function)
The inverse functions of the trigonometric functions , are differentiable with
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
is hence differentiable, and now for there is:
And finally, we determine the derivatives of the hyperbolic functions , and :
Theorem (Derivative of hyperbolic functions)
The functions
are differentiable, and there is
Proof (Derivative of hyperbolic functions)
The derivatives follow directly from the calculation rules. We show only the derivative of . The other two are left to you for practice.
According to the factor and difference rule for all is differentiable, and there is