*Notes to a video lecture on http://www.unizor.com*

__Derivative Properties -__

Linear Combination of Functions

Linear Combination of Functions

Our purpose is to express the derivative of a linear combination of two functions in terms of derivatives of each of them.

Assume that two real functions

**and**

*f(x)***are defined and**

*g(x)**differentiable*(that is, their derivatives exist) on certain interval with derivatives, correspondingly,

*f**and*

^{ I}**(x)**

*g*

^{I}**(x)**Let's determine the derivative of their linear combination

*h(x) = a·f(x)+b·g(x)*where

**and**

*a***are some real numbers.**

*b*The increment of function

**is**

*h(x)*Δ

**=**

*h(x)***Δ**

*h(x+***Δ**

*x)−h(x) =*

= [a·f(x+= [a·f(x+

**Δ**

*x)+b·g(x+***Δ**

*x)] −*

− [a·f(x)+b·g(x)] =

= a·[f(x+− [a·f(x)+b·g(x)] =

= a·[f(x+

**Δ**

*x)−f(x)] +*

+ b·[g(x++ b·[g(x+

**Δ**

*x)−g(x)] =*

= a·= a·

**Δ**

*f(x)+b·*

*g(x)*Next operations to find a derivative are: dividing the function increment Δ

**by an increment of an argument Δ**

*h(x)***and going to a limit as Δ**

*x***.**

*x→0*Δ

**Δ**

*h(x)/***Δ**

*x =*

= [a·= [a·

**Δ**

*f(x)+b·***Δ**

*g(x)]/***Δ**

*x =*

= a·= a·

**Δ**

*f(x)/***Δ**

*x+b·***Δ**

*g(x)/*

*x*Since both functions

**and**

*f(x)***are differentiable, there is a limit of Δ**

*g(x)***Δ**

*f(x)/***and Δ**

*x***Δ**

*g(x)/***as Δ**

*x***. These limits are, correspondingly,**

*x→0*

*f**and*

^{ I}**(x)**

*g*

^{I}**(x)**Recall the properties of limits:

if a sequence has a limit

*L*, then this sequence multiplied by a constant

*k*has limit

*k·L*;

if two sequences have limits

*L*and

*M*, then their sum has a limit

*L+M*.

Therefore,

**Δ**

*a·***Δ**

*f(x)/***→**

*x*

*a·f*

^{ I}**(x)****Δ**

*b·***Δ**

*g(x)/***→**

*x*

*b·g*

^{I}**(x)**and, finally,

Δ

**Δ**

*h(x)/***→**

*x*

*a·f*

^{ I}**(x)+b·g**^{I}**(x)**Hence,

*h*

^{I}**(x) = a·f**^{ I}**(x)+b·g**^{I}**(x)**In other words, derivative of a linear combination of differentiable functions equals to a similar linear combination of their derivatives.

Different forms of notation of this rule are:

(1)

*[a·f(x)+b·g(x)]*

^{ I}**=**

= a·f= a·f

^{ I}**(x)+b·g**^{I}**(x)**(2)

*d*

=

**[a·f(x)+b·g(x)]**/dx ==

**a·**d**f(x)**/dx+**b·**d**g(x)**/dx(3)

*(d/dx)*

=

**[a·f(x)+b·g(x)]**==

**a·**(d/dx)**f(x)+b·**(d/dx)**g(x)**(4)

*D*

=

_{x}**[a·f(x)+b·g(x)]**==

**a·**D_{x}**f(x)+b·**D_{x}**g(x)***Examples*

*[5·sin(x)−7·cos(x)]*

^{ I}**=**

= 5·cos(x)+7·sin(x)= 5·cos(x)+7·sin(x)

*(d/dx)*

**(2x²−3x)**=**4x−3***d*

**(2e**dx =^{x}−x)/**2e**^{x}−1*D*

_{x}**[2x²−3e**=^{x}+4cos(x)]**= 4x−3e**^{x}−4sin(x)
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