Unizor - Derivatives - Linear Combination of Functions

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

Derivative Properties -
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 functionsf(x) and g(x) are defined anddifferentiable (that is, their derivatives exist) on certain interval with derivatives, correspondingly, f ^I(x) and g^I(x).
Let's determine the derivative of their linear combination
h(x) = a·f(x)+b·g(x)
where a and b are some real numbers.

The increment of function h(x)is
Δh(x) = h(x+Δx)−h(x) =
= [a·f(x+Δx)+b·g(x+Δx)] −
− [a·f(x)+b·g(x)] =
= a·[f(x+Δx)−f(x)] +
+ b·[g(x+Δx)−g(x)] =
= a·Δf(x)+b·Δg(x)

Next operations to find a derivative are: dividing the function increment Δh(x) by an increment of an argument Δxand going to a limit as Δx→0.

Δh(x)/Δx =
= [a·Δf(x)+b·Δg(x)]/Δx =
= a·Δf(x)/Δx+b·Δg(x)/Δx

Since both functions f(x) andg(x) are differentiable, there is a limit of Δf(x)/Δx and Δg(x)/Δxas Δx→0. These limits are, correspondingly, f ^I(x) and 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 Land M, then their sum has a limit L+M.

Therefore,
a·Δf(x)/Δx → a·f ^I(x)
b·Δg(x)/Δx → b·g^I(x)
and, finally,
Δh(x)/Δx → 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 ^I(x)+b·g^I(x)
(2) d[a·f(x)+b·g(x)]/dx =
= a·df(x)/dx+b·dg(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_xf(x)+b·D_xg(x)

Examples

[5·sin(x)−7·cos(x)] ^I =
= 5·cos(x)+7·sin(x)

(d/dx)(2x²−3x) = 4x−3

d(2e^x−x)/dx = 2e^x−1

D_x[2x²−3e^x+4cos(x)] =
= 4x−3e^x−4sin(x)