Please Help me derive the derivative of the absolute value of x using the following limit definition.
$$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
$$
I have no idea as to how to get started.Please Help.
Thank You
calculusderivativeslimits
Please Help me derive the derivative of the absolute value of x using the following limit definition.
$$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}
$$
I have no idea as to how to get started.Please Help.
Thank You
Best Answer
$\dfrac{d}{dx}|x|$
$=\lim\limits_{\Delta x\to 0}\dfrac{|x+\Delta x|-|x|}{\Delta x}$
$=\lim\limits_{\Delta x\to 0}\dfrac{(|x+\Delta x|-|x|)(|x+\Delta x|+|x|)}{\Delta x(|x+\Delta x|+|x|)}$
$=\lim\limits_{\Delta x\to 0}\dfrac{|x+\Delta x|^2-|x|^2}{\Delta x(|x+\Delta x|+|x|)}$
$=\lim\limits_{\Delta x\to 0}\dfrac{(x+\Delta x)^2-x^2}{\Delta x(|x+\Delta x|+|x|)}$
$=\lim\limits_{\Delta x\to 0}\dfrac{x^2+2x\Delta x+(\Delta x)^2-x^2}{\Delta x(|x+\Delta x|+|x|)}$
$=\lim\limits_{\Delta x\to 0}\dfrac{2x+\Delta x}{|x+\Delta x|+|x|}$
$=\dfrac{x}{|x|}$
$=\dfrac{x|x|}{|x|^2}$
$=\dfrac{x|x|}{x^2}$
$=\dfrac{|x|}{x}$