Recall the definition of absolute value.
![|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}](https://tex.z-dn.net/?f=%7Cx%7C%20%3D%20%5Cbegin%7Bcases%7D%20x%20%26%20%5Ctext%7Bif%20%7D%20x%20%5Cge%200%20%5C%5C%20-x%20%26%20%5Ctext%7Bif%20%7D%20x%20%3C%200%20%5Cend%7Bcases%7D)
When
,
![|x| = x \implies \dfrac{d|x|}{dx} = 1](https://tex.z-dn.net/?f=%7Cx%7C%20%3D%20x%20%5Cimplies%20%5Cdfrac%7Bd%7Cx%7C%7D%7Bdx%7D%20%3D%201)
When
,
![|x| = -x \implies \dfrac{d|x|}{dx} = -1](https://tex.z-dn.net/?f=%7Cx%7C%20%3D%20-x%20%5Cimplies%20%5Cdfrac%7Bd%7Cx%7C%7D%7Bdx%7D%20%3D%20-1)
The derivative does not exist at
, since the one-sided limits
![\displaystyle \lim_{x\to0^-} f'(x) = -1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%5E-%7D%20f%27%28x%29%20%3D%20-1)
and
![\displaystyle \lim_{x\to0^+} f'(x) = +1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%5Cto0%5E%2B%7D%20f%27%28x%29%20%3D%20%2B1)
do not match.
So the derivative of
is
![\dfrac{d|x|}{dx} = \begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \\ \text{undefined} & \text{if } x = 0\end{cases}](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7Cx%7C%7D%7Bdx%7D%20%3D%20%5Cbegin%7Bcases%7D%201%20%26%20%5Ctext%7Bif%20%7D%20x%20%3E%200%20%5C%5C%20-1%20%26%20%5Ctext%7Bif%20%7D%20x%20%3C%200%20%5C%5C%20%5Ctext%7Bundefined%7D%20%26%20%5Ctext%7Bif%20%7D%20x%20%3D%200%5Cend%7Bcases%7D)
Now we can write this as
![\dfrac{d|x|}{dx} = \dfrac x{|x|} = \dfrac{|x|}x](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7Cx%7C%7D%7Bdx%7D%20%3D%20%5Cdfrac%20x%7B%7Cx%7C%7D%20%3D%20%5Cdfrac%7B%7Cx%7C%7Dx)
since
![x > 0 \implies |x| = x \implies \dfrac{|x|}x = \dfrac xx = 1](https://tex.z-dn.net/?f=x%20%3E%200%20%5Cimplies%20%7Cx%7C%20%3D%20x%20%5Cimplies%20%5Cdfrac%7B%7Cx%7C%7Dx%20%3D%20%5Cdfrac%20xx%20%3D%201)
and
![x < 0 \implies |x| = -x \implies \dfrac{|x|}x = -\dfrac xx = -1](https://tex.z-dn.net/?f=x%20%3C%200%20%5Cimplies%20%7Cx%7C%20%3D%20-x%20%5Cimplies%20%5Cdfrac%7B%7Cx%7C%7Dx%20%3D%20-%5Cdfrac%20xx%20%3D%20-1)
Answer:
3.183.10^7
Step-by-step explanation:
Each pair has 2 socks so 19÷2= 9 with 1 remaining unmatched sock.
Answer:
2 dollars each I think........
Answer:
J. 500 inches
Step-by-step explanation:
There are 4 sides to a square so, to find the measure of each of the lengths, u have to... 2,000 ÷ 4 = 500 inches each side length