Saturday, October 8, 2016

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,
(xx)n=xn+n·xn−1·Δx+P·Δ2x
where P is some polynomial of x and Δx.
Therefore,
[(xx)nxn]/Δx =
n·xn−1+P·Δx
As Δx0, 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 √xx − √x
by √xx + √x
Since (a−b)·(a+b)=a²−b², the result will be
Δx/(xx + √x)
Dividing this by increment of argument Δx and going to the limit as Δx0, the result will be 1/(2√x)

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