Computing derivatives - special – Serlo

Special cases of the chain rule

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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).

Case:   is linear

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Let   and let   be differentiable. Then also   is differentiable ad at   there is

 

Proof

  is differentiable with   for all  . The chain rule implies differentiablilty of   , where

 

Example

Let   with  . Then   for all   so

 

Case:   is a power function

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Let   be differentiable. The also   is differentiable for all  , where at   there is

 

Proof

  is differentiable with   for all  . The chain rule implies differentiablility of  , where

 

Example (Deriving a power function)

Let   with  . Then   and for all  we have

 

Case:   is a root function

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Let   be differentiable. then   with   is differentiable as well and for all   there is

 

Proof

  is differentiable with   for all  . The chain rule implies differentiability of   where

 

Example (Deriving a root function)

Let   with  . Then   and for all   there is

 

Case:  

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Let   be differentiable. Then   is differentiable as well and for all   there is

 

Proof

Let   , which is differentiable with  . Since   is differentiable by assumption, we also get differentiability of  . By the chain rule,

 

Example (Deriving exponential functions)

1. Let   with  . Then,   for all   and we have

 

2. Let   with  . Then,   for all   and we have

 

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

 

Case:  

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Let   be and by the chain rule. Then,   is and by the chain rule as well and for all   there is

  (logarithmic derivative)

Proof

Let   , which is and by the chain rule with   for all  . Since   is and by the chain rule by assumption, the chain rule implies differentiability of   and

 

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:

  1. Why is the domain of   only  ?
  2. What of domain of  ?

Solutions:

  1. There is   is well-defined
  2. There is   is well-defined. So  . For the derivative of   there is
 

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. ( )

Linear combinations of functions

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The factor and sum rule state that the derivative is linear. If we apply this linearity to   functions, we get:

Theorem (Differentiating linear combinations of functions)

Let  ,   be differentiable and  . Then,

 

is differentiable as well and for all   there is

 

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

 

Application: Deriving sum formulas

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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  ?

For   we get

 

and for  

 

Generalized product rule

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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:

Theorem (Generalized product rule)

Let   and let   be differentiable. The the product function   is also differentiable with

 

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:  .

 

Example (Generalized product rule)

The function

 

is differentiable, since  ,   and   are differentiable for all   . In addition

 ,   and  

The generalized product rule yields for  :

 

Exercise (Generalized product rule)

Determine the domain of definition and the derivative of

 

Solution (Generalized product rule)

Domain of definition: The functions  ,   and   are defined on all of  . by contrast,   is only defined on  . Hence

 

Derivative:   is differentiable, as the functions  ,  ,   and   are differentiable. In addition, for all   there is:

 ,  ,   and  

The generalized product rule yields:

 

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.

Logarithmic derivatives

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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:

    Domain of definition
     
 ,      
     
     
     
     
     

Exercise (Computing logarithmic derivatives)

Determine the logarithmic derivative (with domain of definition) of the following functions

  1.  
  2.  
  3.   with  

Solution (Computing logarithmic derivatives)

Part 1: There is   for all  . So

 

Since   for all  , the domain of definition for our logarithmic derivative of   is equal to  .

Part 2: The quotient rule yields

  for all  

So

 

Since   for all  , the domain of definition for our logarithmic derivative of   is equal to  .

Part 3: For   there is

 

Since   for all  , the domain of definition for our logarithmic derivative of   is equal to  .

By direct computation we obtain the following rules for the logarithmic derivative:

Theorem (Computation rules for logarithmic derivatives)

For two differentiable functions   and   without zeros, there is

  1.  
  2.  
  3.  
  4.   for  
  5.   for  

Note: The rules are analogous to the computation rules for the logarithm function.

Proof (Computation rules for logarithmic derivatives)

We will only prove rule 1 and rule 4, the other three we leave to you "as an exercise" (don't worry, there is a solution, here).

Rule 1: Since   and   are differentiable and free of zeros,   is also differentiable and free of zeros. Thus the following applies

 

Rule 4: Since   is differentiable and zero-free,   is also differentiable and zero-free for  . Further above we have already shown   by the chain rule. So

 

Exercise (Computation rules for logarithmic derivatives)

Prove rules 2, 3 and 5 of the previous theorem

Proof (Computation rules for logarithmic derivatives)

Rule 2: Since   are differentiable and zero-free,   is also differentiable and zero-free. By the chain rule,  . Thus, there is

 

Rule 3: Since   and   are differentiable and zero-free,   is also differentiable and zero-free. Using rules 1 and 2 we get

 

Alternatively, the rule can be proved by using the quotient rule.

Rule 5: Since   are differentiable and positive,   is also differentiable and positive. With the chain rule,   applies. Thus,

 

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!

Example (Logarithmic derivatives 1)

First we differentiate the following product function by means of the logarithmic derivation

 

First we determine the domain of definition: there is  ,   and  . In order to be able to form the logarithmic derivative,   and   must be zero-free. Because of   we will choose  .


Now take the logarithmic derivative of  : There is

 

Finally, we multiply the equation by   and obtain

 

Example (Logarithmic derivatives 2)

Next, we differentiate the following quotient function:

 

Concerning the domain: The denominator is always different from zero. In order for   to be free of zeros, the numerator must be unequal to zero. Therefore:

 

Hence, the domain of definition is  .

With   and   we have for the logarithmic derivative of  :

 

Multiplication by   yields:

 

Example (Logarithmic derivatives 3)

Finally, we differentiate with the logarithmic derivative

 

Concerning the domain: In order for   to be defined,   must hold. The function   is zero-free on all of  . So  .

The logarithmic derivative of   is

 

Multiplication by   yields:

 

Exercise (Logarithmic derivatives)

Using the logarithmic derivatives, differentiate the following functions on their domain of definition:

  1.  
  2.  
  3.  

Generalized chain rule

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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