# Check if the following functions are differentiable

## Homework Statement

Check if the following functions f : ℝ → ℝ are differentiable:
$\displaystyle f(x)=|(x-1)^{2}(x+1)^{3}|$

$\displaystyle f(x)=|x^{2}-\pi^{2}|sin^{2}x$

## The Attempt at a Solution

I don't know what the condition should be, I've searched a lot of topics concerning this problem but they all contain reference to some points or intervals at which it should be checked.
That's why I have no idea how to start and what definition has to be used.

Simon Bridge
Homework Helper
Do you know what a derivative is?
What it means for the derivative to exist at a point?

Notice that these are absolute value functions.
What does that do to the values?
Have you tried plotting any of them?

They are "obviously" differentiable, except at a set of isolated points. Check those points. Show that the limit at x (from the definition of the derivative) does not exist. You can show this by noting that if a limit exists, it is unique.

They are "obviously" differentiable, except at a set of isolated points. Check those points. Show that the limit at x (from the definition of the derivative) does not exist. You can show this by noting that if a limit exists, it is unique.

After hours of reading I finally got to that, but the problem now is how to divide $\displaystyle f(x)=|(x-1)^{2}(x+1)^{3}|$ into cases so that it'd be so easily calculated as eg. $|x-2|$ where we have 2 cases - for x > 2 and x < 2, then make the derivative of both functions and check whether it's the same or not.

If you square something, it's always positive, so you can remove the absolute value sign.

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

Check if the following functions f : ℝ → ℝ are differentiable:
$\displaystyle f(x)=|(x-1)^{2}(x+1)^{3}|$

$\displaystyle f(x)=|x^{2}-\pi^{2}|\sin^{2}x$

## The Attempt at a Solution

I don't know what the condition should be, I've searched a lot of topics concerning this problem but they all contain reference to some points or intervals at which it should be checked.
That's why I have no idea how to start and what definition has to be used.

Hello Pole . Welcome to PF !

Have you learned about the differentiability of any broad classes of functions, such as polynomials, sinusoidal (sin(x) & cos(x)) functions, exponential functions, etc. ?

The absolute value function, |x|, has a piecewise definition, so when it is combined with other functions, you should check at those places at which the function is 'pieced together' .

SammyS
Staff Emeritus
$f(x)=(x-1)^2\left|(x+1)^3\right|$​
$\displaystyle |x|=\left\{\matrix{x\,,\text{ if }\ x\ge0\\ -x\,,\text{ if }\ x<0} \right.$​