Find the derivative of the function $ y= x|\cos{\frac{\pi}{x}}|$

Asked Feb 02 '16 at 14:56

Active Feb 02 '16 at 14:58

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Function is defined as it follows : $x \neq 0$ and $f(0)=0$

My work is:

$\frac{d}{dx}(x|\cos{\frac{\pi}{x}}|)$ = $|\cos{\frac{\pi}{x}}|$ + $x(\frac{d}{dx}|\cos{\frac{\pi}{x}}|)$ = $|cos{\frac{\pi}{x}}|$ + $\frac{\pi}{x}(\operatorname{sgn}(\cos{\frac{\pi}{x}}))(\sin{\frac{\pi}{x}})$ = $\operatorname{sgn}(\cos{\frac{\pi}{x}})[(\cos{\frac{\pi}{x}}) + \frac{\pi}{x}(\sin{\frac{\pi}{x}})]$

Should I consider any points on the domain and find the derivative of the function on those points?

Thank you

edited Feb 02 '16 at 14:58

Giovanni

6,321

asked Feb 02 '16 at 14:56

Premtim

2

The derivative at $x=0$ needs to be calculated separately. – GoodDeeds Feb 02 '16 at 15:01
@GoodDeeds i can't see the mistake in my calculation? Are there any other points to be mentioned? – Premtim Feb 02 '16 at 15:14
But there is a $x$ multiplying that expression. – Premtim Feb 02 '16 at 15:25
Yes you are right that was careless of me, sorry. – GoodDeeds Feb 02 '16 at 15:26
Just wondering as an alternative, would it be easier to write that absolute value term as the square root of its square and then taking a derivative? – imranfat Feb 02 '16 at 15:26
You need to take care of $x_0=0$ and of those points where $\cos (\pi/x)=0$ separately, since the modulus is not differentiable at $0$. – PhoemueX Feb 02 '16 at 17:21
$x=0$ is an essential singularity. – Ng Chung Tak Feb 02 '16 at 19:26
@TheGreatDuck: The absolute value function $x \mapsto |x|$. – PhoemueX Jun 23 '16 at 21:09
@TheGreatDuck: Because everyone is doing this: https://en.m.wikipedia.org/wiki/Modulus – PhoemueX Jun 24 '16 at 07:18

Find the derivative of the function $ y= x|\cos{\frac{\pi}{x}}|$

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