# How to differentiate an absolut value, f(x)=│x^2-4│

• gillgill
In summary, the differentiation of the function, f(x), is differentiable at x=2 and x=-2 if and only if the left and right hand derivatives exist at those points.
gillgill
how do u differentiate f(x)=│x^2-4│...?
i don't know how to do it with absolute values...

If I remember right, define it in a piecewise fashion. Can you see that:

|x^2 - 4| = x^2 - 4 if x^2 - 4 > 0

|x^2 - 4| = -(x^2 - 4) if x^2 - 4 < 0

The first condition becomes: if x^2 > 4 ===> |x| > 2, i.e. if x > 2 OR x < -2

The second becomes: if x^2 < 4 ===> |x| < 2, i.e. -2 < x < 2

So you have two cases. For x > 2 and x < -2, the function is:

f(x) = x^2 - 4, and you can differentiate it.

For -2 < x < 2, the function is:

f(x) = -x^2 + 4, and you can differentiate it.

The function is differentiable at x = 2 and x = -2 if and only if the left and right hand derivatives exist at those points.

It is quite instructive to use the DEFINITION of the derivative at the problem points 2 and -2.
I'll take the "2"-case:
In general, we have:
$$f'(2)=\lim_{\bigtriangleup{x}\to{0}}\frac{f(2+\bigtriangleup{x})-f(2)}{\bigtriangleup{x}}$$
if it exists.
In our case, $$f(x)=|x^{2}-4|$$
which implies $$f(2)=0,f(2+\bigtriangleup{x})=|(2+\bigtriangleup{x})^{2}-4|=|4\bigtriangleup{x}+(\bigtriangleup{x})^{2}|$$
Hence, we must have:
$$f'(2)=\lim_{\bigtriangleup{x}\to0}\frac{|\bigtriangleup{x}|}{\bigtriangleup{x}}|4+\bigtriangleup{x}|$$
if it exists.
Can it exist?

i found the first reply easier to understand...
f(x)=x^2-4
f'(x)=2x
f'(x) as x->2+ would equal 2(2)=4

f(x)=-x^2+4
f'(x)=-2x
f'(x) as x->2- would equal -2(2)=-4
so does not exist at x=2

f(x)=x^2-4
f'(x)=2x
f'(x) as x->-2- would equal 2(-2)=-4

f(x)=-x^2+4
f'(x)=-2x
f'(x) as x->-2+ would equal -2(-2)=4
so does not exist at x=-2
is that right?

Yes that's right, because the limit does not exist therefore the derivative does not exist at those points.

okay...i see...thanks guys...

## What is an absolute value?

An absolute value represents the distance of a number from zero on a number line, regardless of its sign. It is always positive or zero.

## How do I find the absolute value of an expression?

To find the absolute value of an expression, you can use the absolute value symbol, which is two vertical bars around the expression. For example, the absolute value of -3 would be written as | -3 | and would equal 3.

## What is the difference between an absolute value and a regular value?

The main difference is that an absolute value is always positive or zero, while a regular value can be positive, negative, or zero. Absolute values are also represented by two vertical bars, while regular values are not.

## How do I differentiate an absolute value function?

In order to differentiate an absolute value function, you need to use the chain rule. The derivative of an absolute value function is equal to the derivative of the inside function multiplied by the sign of the inside function. For example, the derivative of |x| would be equal to 1 if x is positive and -1 if x is negative.

## What is the derivative of f(x)=│x^2-4│?

The derivative of f(x)=│x^2-4│ is equal to 2x if x is positive and -2x if x is negative. This can also be written as 2x/|x|.

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