Differentiation help

[MickyMouse!]

Hey guys, was just wondering if someone could help me differentiate the following function:

f(x)= Absolute sin(4x)

Also, Is the function f(x) differentiable at pi/4?

Differentiation of circular functions is my worst area in the topic so any help would be appreciated.

[Homer]

as far as i know, to differentiate an absolute value function, you would need to differentiate both f(x) and -f(x) and create a hybrid function of f'(x) with appropriate intervals. Normally the graph is not differentiable at cusps values hence no you cannot differentiate at pi/4

[Phy124]

Example over the domain :





The function does not have a defined derivative at . For VCE purposes the explanation of their being a sharp change in gradient at these points would suffice.


Alternatively,

for hence the denominator is zero for these values and consequently is undefined.

[lzxnl]

The derivative of |x| is |x|/x. Try it. It works.

f(x) = |sin 4x|
f'(x) by chain rule is 4cos 4x |sin 4x|/sin 4x

Example over the domain :





The function does not have a defined derivative at . For VCE purposes the explanation of their being a sharp change in gradient at these points would suffice.


Alternatively,

for hence the denominator is zero for these values and consequently is undefined.

I think it would be slightly better to say that as x approaches pi*n/4 from either direction, the limit is different, so it's undefined. Derivatives are themselves defined by a limit, so in the working out you may well end up with your relation on the last row and having to take the limit as x => pi/4
It's like saying that the derivative of sin x at x=0 is undefined as you have to take the limit of sin x/x as x approaches 0 but the bottom is 0. Zero denominators, as you are probably very familiar with, don't always mean undefined limits.

[Phy124]

The derivative of |x| is |x|/x. Try it. It works.

See this post if anyone struggles to find the relation.

Spoiler

I'm almost certain that you can't cancel out the ds like that. It's essentially the same as the sin x/n=6 thing.

Think of it like this;















Sub back in



This gives us (as we worked out differently before):





(Just realised how long it's been since I've done the full working out for differentiating, so much effort )

I think it would be slightly better to say that as x approaches pi*n/4 from either direction, the limit is different, so it's undefined. Derivatives are themselves defined by a limit, so in the working out you may well end up with your relation on the last row and having to take the limit as x => pi/4
It's like saying that the derivative of sin x at x=0 is undefined as you have to take the limit of sin x/x as x approaches 0 but the bottom is 0. Zero denominators, as you are probably very familiar with, don't always mean undefined limits.

Yes, it's certainly better to say what you have. But for VCE purposes (math methods in particular, I can't really speak for spesh), where limits are barely touched on, I feel it is sufficient to know that there will be no derivative when;
- there is a sharp corner
- there is a vertical inflection point
- there is a point of discontinuity (infinite, removable or jump)

I know it is important to have a wealth of knowledge and be able to comprehend the underlying principles, but for things like this I've found that with the students I have taught simplifying the material and being familiar with the above three points has yielded the best results.

Each student has their own preferences, though. So to all students, whatever you think will get you the best results, do that

[lzxnl]

I recall that the Methods textbook has a definition of differentability as stating a function is differentiable at a point if the limit of the difference quotient exists (and of course if the function is defined at that point), and the textbook also makes mention of when a limit is defined, so putting these two together, it should be within the VCE course to give a limit definition of differentability and looking at these limits from a left and right hand side perspective.

[b^3]

Just because certain things are in the textbook, doesn't mean that they're on the course Although limits and such are, they are not done well. They're skimmed over mostly, which is unfortunate. You hardly get questions that are much harder than finding the limit of a polynomial by just substituting in the value, on VCAA exams.

Looking for the cusps isn't wrong either, in the study design in the advice for teachers section, they encourage to look at it graphically.

Power functions and generalisation to rational powers on positive real domain. Review of
transformations in the context of power functions. Introduction of modulus function, and
transformations of this function, as an example of a function that is continuous, but not
differentiable at a given point (graphical treatment).

pg 142 - http://www.vcaa.vic.edu.au/Documents/vce/mathematics/mathsstd.pdf

In other words, if you want to use limits, use them, if you want to stick to what 2/cos(c) has said, that's fine too. It's just another example of where VCE maths doesn't do everything the way it should be doing it.