Now we want to list a few special cases of the chain rule, which occur frequently in practice. For the derivation of the derivatives of , , , , etc. we refer to the following chapter Examples for derivatives (missing).
Let and let be differentiable. Then also is differentiable ad at there is
Case: is a power function
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Let be differentiable. The also is differentiable for all , where at there is
Case: is a root function
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Let be differentiable. then with is differentiable as well and for all there is
Case:
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Let be differentiable. Then is differentiable as well and for all there is
Special case: Differentiating "function to the power of a function"
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Consider the function
which is a special case of an exponential function. The inner function is . We may again just use the chain rule.
Example (Deriving exponential functions 2)
1. Let with . Then, for all and by the chain rule
2. Let with . Then, for all and by the chain rule
3. Let with . Then, for all and by the chain rule
Let be and by the chain rule. Then, is and by the chain rule as well and for all there is
(logarithmic derivative)
Example (Logarithmic derivatives)
1. Let with . Then, for all and by the chain rule
2. Let with . Then, for all and by the chain rule
Questions for understanding: Answer the questions:
- Why is the domain of only ?
- What of domain of ?
Hint
Below we will see how we can use the logarithmic derivative to calculate easily the derivatives of product, quotient or power functions. This makes sense especially if the function consists of several products, for example. ()
The factor and sum rule state that the derivative is linear. If we apply this linearity to functions, we get:
Proof (Differentiating linear combinations of functions)
We show the assertion by induction over :
Induction base: . For there is
Induction assumption:
shall hold for some
Induction step: .
Example (Differentiability of polynomial functions)
The power function is differentiable for all where
The theorem from above applied to polynomial functions yields
for and differentiable with
We can use the linearity of the derivative to obtain new sum formulas from already known ones. Let us consider as an example the geometric sum formula (missing) for and :
Both sides of the equation can be understood as differentiable functions or or :
Since is a polynomial, we have for :
Furthermore, by the quotient rule
Since now , we also have . So for there is:
Additional question: Which sum formulas do we get for and ?
The product rule can also be applied to more than two differentiable functions by first combining several functions and then applying the product rule several times in succession. For three functions we get
For four functions we get analogously
We now recognize a clear formation law for derivatives: the product of the functions is added up, whereby in each summand the derivative "moves forward" by one position. In general, the derivative of a product function of functions is:
Exercise (Proof of the generalized product rule)
Prove the generalized product rule by induction over .
Proof (Proof of the generalized product rule)
Induction base: . Es gilt
Induction assumption:
is assumed for some
Induction step: .
Exercise (Generalized product rule)
Determine the domain of definition and the derivative of
Hint
If additionally for all , we can divide both sides by this product, and thus obtain the form
The advantage of this representation is that the sum on the right side is much clearer. This is already the idea behind the logarithmic derivative, which we present in the next section.
The logarithmic derivative is a very elegant tool to calculate the derivative of some functions of a special form. For a differentiable function without zeros, the logarithmic derivative is defined by
We have already shown above that the chain rule yields:
The following table lists some standard examples of logarithmic derivatives:
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Domain of definition
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By direct computation we obtain the following rules for the logarithmic derivative:
Note: The rules are analogous to the computation rules for the logarithm function.
Exercise (Computation rules for logarithmic derivatives)
Prove rules 2, 3 and 5 of the previous theorem
Hint
The summation rule can still be generalized to zero-free and differentiable ( as
Using those rules, we can now easily calculate derivatives. The transition to logarithmic derivatives does not usually require less computational effort, but it is much clearer than calculating with the usual rules, and therefore less susceptible to errors!
Exercise (Logarithmic derivatives)
Using the logarithmic derivatives, differentiate the following functions on their domain of definition:
Just like the sum and product rule, the chain rule can be generalized to the composition of more than two functions. For two differentiable functions and the chain rule reads
If we have three functions , and , then by applying the rule twice we obtain
If we now take a closer look, we can see a law of formation: First the outermost function is differentiated and the two inner ones are inserted into the derivative function. Then the second function is differentiated and the innermost function is inserted, and the whole thing is multiplied by the first derivative. Finally, the innermost function is differentiated and multiplied. If we now generalize this to functions, we get:
Theorem (Generalized chain rule)
Let be differentiable for all , and for all . Then is also differentiable, and its derivative at is given by
Proof (Generalized chain rule)
We prove the theorem by induction over :
Induction base: . There is
. The chain rule yields
Induction assumption:
is assumed for all
and some
Induction step: . For there is