Linearity of differentiation

In calculus, the derivative of any linear combination of functions equals the same linear combination of the derivatives of the functions;[1] this property is known as linearity of differentiation, the rule of linearity,[2] or the superposition rule for differentiation.[3] It is a fundamental property of the derivative that encapsulates in a single rule two simpler rules of differentiation, the sum rule (the derivative of the sum of two functions is the sum of the derivatives) and the constant factor rule (the derivative of a constant multiple of a function is the same constant multiple of the derivative).[4][5] Thus it can be said that differentiation is linear, or the differential operator is a linear operator.[6]

Statement and derivation

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Let f and g be functions, with α and β constants. Now consider

 

By the sum rule in differentiation, this is

 

and by the constant factor rule in differentiation, this reduces to

 

Therefore,

 

Omitting the brackets, this is often written as:

 

Detailed proofs/derivations from definition

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We can prove the entire linearity principle at once, or, we can prove the individual steps (of constant factor and adding) individually. Here, both will be shown.

Proving linearity directly also proves the constant factor rule, the sum rule, and the difference rule as special cases. The sum rule is obtained by setting both constant coefficients to  . The difference rule is obtained by setting the first constant coefficient to   and the second constant coefficient to  . The constant factor rule is obtained by setting either the second constant coefficient or the second function to  . (From a technical standpoint, the domain of the second function must also be considered - one way to avoid issues is setting the second function equal to the first function and the second constant coefficient equal to  . One could also define both the second constant coefficient and the second function to be 0, where the domain of the second function is a superset of the first function, among other possibilities.)

On the contrary, if we first prove the constant factor rule and the sum rule, we can prove linearity and the difference rule. Proving linearity is done by defining the first and second functions as being two other functions being multiplied by constant coefficients. Then, as shown in the derivation from the previous section, we can first use the sum law while differentiation, and then use the constant factor rule, which will reach our conclusion for linearity. In order to prove the difference rule, the second function can be redefined as another function multiplied by the constant coefficient of  . This would, when simplified, give us the difference rule for differentiation.

In the proofs/derivations below,[7][8] the coefficients   are used; they correspond to the coefficients   above.

Linearity (directly)

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Let  . Let   be functions. Let   be a function, where   is defined only where   and   are both defined. (In other words, the domain of   is the intersection of the domains of   and  .) Let   be in the domain of  . Let  .

We want to prove that  .

By definition, we can see that

 


In order to use the limits law for the sum of limits, we need to know that   and   both individually exist. For these smaller limits, we need to know that   and   both individually exist to use the coefficient law for limits. By definition,   and  . So, if we know that   and   both exist, we will know that   and   both individually exist. This allows us to use the coefficient law for limits to write

 

and

 

With this, we can go back to apply the limit law for the sum of limits, since we know that   and   both individually exist. From here, we can directly go back to the derivative we were working on. Finally, we have shown what we claimed in the beginning:  .

Let   be functions. Let   be a function, where   is defined only where   and   are both defined. (In other words, the domain of   is the intersection of the domains of   and  .) Let   be in the domain of  . Let  .

We want to prove that  .

By definition, we can see that

 In order to use the law for the sum of limits here, we need to show that the individual limits,   and   both exist. By definition,  and  , so the limits exist whenever the derivatives   and   exist. So, assuming that the derivatives exist, we can continue the above derivation

 


Thus, we have shown what we wanted to show, that:  .

Difference

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Let   be functions. Let   be a function, where   is defined only where   and   are both defined. (In other words, the domain of   is the intersection of the domains of   and  .) Let   be in the domain of  . Let  .

We want to prove that  .

By definition, we can see that:  

In order to use the law for the difference of limits here, we need to show that the individual limits,   and   both exist. By definition,   and that  , so these limits exist whenever the derivatives   and   exist. So, assuming that the derivatives exist, we can continue the above derivation

 

Thus, we have shown what we wanted to show, that:  .

Constant coefficient

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Let   be a function. Let  ;   will be the constant coefficient. Let   be a function, where j is defined only where   is defined. (In other words, the domain of   is equal to the domain of  .) Let   be in the domain of  . Let  .

We want to prove that  .

By definition, we can see that:

 

Now, in order to use a limit law for constant coefficients to show that

  we need to show that   exists. However,  , by the definition of the derivative. So, if   exists, then   exists.

Thus, if we assume that   exists, we can use the limit law and continue our proof.

 

Thus, we have proven that when  , we have  .

See also

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References

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  1. ^ Blank, Brian E.; Krantz, Steven George (2006), Calculus: Single Variable, Volume 1, Springer, p. 177, ISBN 9781931914598.
  2. ^ Strang, Gilbert (1991), Calculus, Volume 1, SIAM, pp. 71–72, ISBN 9780961408824.
  3. ^ Stroyan, K. D. (2014), Calculus Using Mathematica, Academic Press, p. 89, ISBN 9781483267975.
  4. ^ Estep, Donald (2002), "20.1 Linear Combinations of Functions", Practical Analysis in One Variable, Undergraduate Texts in Mathematics, Springer, pp. 259–260, ISBN 9780387954844.
  5. ^ Zorn, Paul (2010), Understanding Real Analysis, CRC Press, p. 184, ISBN 9781439894323.
  6. ^ Gockenbach, Mark S. (2011), Finite-Dimensional Linear Algebra, Discrete Mathematics and Its Applications, CRC Press, p. 103, ISBN 9781439815649.
  7. ^ "Differentiation Rules". CEMC's Open Courseware. Retrieved 3 May 2022.
  8. ^ Dawkins, Paul. "Proof Of Various Derivative Properties". Paul's Online Notes. Retrieved 3 May 2022.