Unizor - Derivatives - Product of Functions

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

Derivative Properties -
Product of Two Functions

Our purpose is to express the derivative of a product 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 product
h(x) = f(x)·g(x)

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

At the same time
f(x+Δx) → f(x)
as Δx→0.

Recall the properties of limits:
if two sequences have limits Land M, then their sum has a limit L+M and their product has a limit L·M.

Therefore,
Δh(x)/Δx→f(x)·g^I(x)+g(x)·f ^I(x)

Hence,
h^I(x) = f(x)·g^I(x)+g(x)·f ^I(x)

It's called the Product Rule of differentiation.
Different forms of notation of this rule are:
(1) [f(x)·g(x)] ^I =
= f ^I(x)·g(x)+f(x)·g^I(x)
(2) d[f(x)·g(x)]/dx =
= f(x)·dg(x)/dx+g(x)·df(x)/dx

Examples

[x³·cos(x)] ^I =
= 3x²·cos(x)−x³·sin(x)

d(3x³·2^x)/dx =
= 3x³·ln(2)·2^x+9x²·2^x

(d/dx)[sin(2x)] =
= (d/dx)[2sin(x)·cos(x)] =
= 2cos(x)·cos(x)−2sin(x)·sin(x)
= 2[cos²(x)−sin²(x)] =
= 2cos(2x)

D_x[x³·5^x·cos(x)] =
= D_x[(x³·5^x)·cos(x)] =
= D_x(x³·5^x)·cos(x) + (x³·5^x)·D_x[cos(x)] =
= D_x(x³)·5^x·cos(x) + x³·D_x(5^x)·cos(x) + x³·5^x·D_x[cos(x)] =
= 3x²·5^x·cos(x) + x³·ln(5)·5^x·cos(x) − x³·5^x·sin(x)