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On the page about partial differentiation f is a function of y and x. So if you wanted to make a graph of this function a sheet of 2 dimensional graph paper wouldn't be enough. You need a 3 dimensional graph with f having its values plotted in the 'z' direction. Imagine you have the x and y axes on a flat surface and the z axis goes up in the air. If you choose values of x and y, there will be values of z making a (wavy) surface above (and below) the x-y plane.

For partial differentiation we assume that one of the variables (let's say y) is fixed. That means we are only considering a part of the surface where x and z can vary but y stay constant. This is now a 2 dimensional problem once more so we can use the usual rules of differentiation. Let's imagine a possible situation where we could use this. If the surface has a dip down to a lowest point, taking y as constant will be like taking a slice through the surface and making a curve which has a lowest point. If we now re-do the differentiation with x constant there will similarly be a lowest point. For each there will still be a variable (the one that we've been pretending is constant) in the equations. But with a pair of such equations we can find the single point where both x and y are lowest together, in other words the bottom of the dip.

In the same way we can find maximum points on the surface and also saddle points where one variable is lowest and the other highest. If you are walking towards a mountain pass, that is an example of a saddle point. As you go through the pass the ground slopes down in front and behind so you are at a maximum in that direction; but to either side the mountain sides slope up, so in that direction you are at a minimum. In both directions the partial derivatives are zero.

Now back to the 2 dimensional cases. If the function is expressed as y = function of x, then the differentiation is relatively straight forward. But sometimes we don't have that; we may have an equation such as x^2 + y^2 = R where neither variable is the subject of the equation. In this case it is possible to re-arrange the equation so that y is the subject, but it's not an easy differentiation then, and sometimes you cannot do the re-arrangement at all.

That's when implicit differentiation becomes useful. Whenever you are differentiating it is always with respect to a variable, let's say x. Do the x containing components as usual but you have to remember that y is a variable too. So it must be differentiated with respect to x as well. And that's the process of implicit differentiation.

It's worth noting that even with a simple function like y = x^2 you are actually using implicit differentiation. The right hand side becomes 2x. The left hand side becomes dy/dx. So it has also been differentiated with respect to x.

Hope that helps,

Bob

Thanks for the good wishes in the other post. Let me know if you discover how to achieve it Living for a 1000 years that is.

]]>Thank you bob. May you live for thousand years ]]>

Bob

]]>I have been tearing my hair out over this lol.

I am trying to work my holiday (vacation) entitlement out. Can somebody please tell me if I am right?

I get 27 days holiday plus the bank holidays (TroiaResort holidays). My work year is 1st of April to 31st of March. Due to Easter moving about that means this year I get 7 days as bank holidays. In total this year I get 34 days holiday entitlement

If I worked full time I would do 7.5 hours per day and work 5 days per week = 37.5 hours per week. 34 holidays days x 7.5 hours a day = 255 hours holiday entitlement. If I was full time every time I take holiday day I would lose 7.5 hours/1 day from my holiday entitlement.

I don’t work full time. I do 18.45 hours per week which is half a week’s work. I don’t work 2.5 days though. I work 3 days per week and do 6.45 hours a day.

Based on that information am I correct in saying

This year I get 127.5 hours holiday entitlement (half of a full time worker). Every time I take a holiday day I take off 6.15 hours from my holiday entitlement

Or should it simply be

I get 34 days holiday entitlement. Every time I take a holiday day I just take a day off my holiday entitlement.

I think you have calculated right not required any changes.

]]>I think your thread was taken from my thread with the same

name posted 2 years ago .

This problem had already been solved with the help of

bob bundy , you can search the thread and find the solution .

Still thankful to bob bundy !

]]>Angles PQS and SQR are shown with two arcs. I think this can be interpreted as meaning that they are equal.

So in triangle PQS that angle can be found and then in triangle SQR you know two (and therefore all three) angles and a side. So x can be found.

Bob

]]>Welcome to the forum.

In mechanics, to switch between acceleration, a, velocity, v and distance, s it is necessary to differentiate and integrate as

I seem to remember this bit of algebraic manipulation arises when dealing with simple harmonic motion, because a is defined as a function of s rather than t. So a way is needed to express a in terms that can be integrated.

If you post back more precisely where you've got to in your course and why you came searching here, I'll try to provide more help.

Bob

ps. s is used for distance as d is likely to be used for differentiating. x may also be used.

]]>ANSWER:

(500+1800+750)=$3050

(3050/180)=$16.94 per month

($16.94 x 6 months)= $101.64 Can be deducted in current year as ( the expenses for the sale of stock are not organizational expenditures)

and hence

which is the required solution. Your solution is also correct, but it required some manipulation to reach the desired form.]]>

Thank you all so much for taking the time to help out. I do appreciate. It appears there is no solution for the question. There must have been an error in whatever book from which my mate sent me that question.

Regards,

math9maniac

Welcome to the forum.

This problem came up ages ago so I've mostly forgotten it. Some other posters provided quicker solutions than mine; that's fine; I never said mine was the only one nor that it was quick. The OP was happy to be taught some new maths along the way which I was happy to do.

PQRC is a sloping plane. So QR isn't a continuation of PQ. Not sure what you mean here.

Bob

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