Saturday, October 8, 2016
Unizor - Derivatives - Examples - Power Function
Notes to a video lecture on http://www.unizor.com
Derivative Examples -
1. f(x) = xn
f'(x) = n·xn−1
Using Newton's binomial formula,
where P is some polynomial of x and 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.
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)