Unizor - Derivatives - Examples - Power Function

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

Derivative Examples -
Power Functions

1. f(x) = xⁿ

f'(x) = n·xⁿ⁻¹

Proof
Using Newton's binomial formula,
(x+Δx)ⁿ=xⁿ+n·xⁿ⁻¹·Δx+P·Δ²x
where P is some polynomial of x and Δx.
Therefore,
[(x+Δx)ⁿ−xⁿ]/Δx =
= n·xⁿ⁻¹+P·Δx
As Δx→0, this expression tends to n·xⁿ⁻¹


2. f(x) = √x (defined for x ≥ 0)

f'(x) = 1/(2√x)
Notice that derivative does not exist for x=0. So, the domain of a derivative is narrower than the domain of the original function.

Proof
We assume that x takes only positive values.
Let's multiply and divide the expression for function increment √x+Δx − √x
by √x+Δx + √x
Since (a−b)·(a+b)=a²−b², the result will be
Δx/(√x+Δx + √x)
Dividing this by increment of argument Δx and going to the limit as Δx→0, the result will be 1/(2√x)