Saturday, October 8, 2016

Unizor - Derivatives - Examples - Power Function

Notes to a video lecture on

Derivative Examples -
Power Functions

1. f(x) = xn

f'(x) = n·xn−1

Using Newton's binomial formula,
where P is some polynomial of x and Δx.
[(xx)nxn]/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.

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)

No comments: