Mastering Differentiation Calculus: Solving Homework Equations with Ease

[toasticles]

Homework Statement

Differentiate

Homework Equations

see 1.

The Attempt at a Solution

btw the x in the second equation is meant to be an r
how much did i fail? :(

 

[sponsoredwalk]

Why is $$\pi$$ in the numerator in the last term?
The mislabelled "r" is just a mistake, let this be a lesson to keep an eye on what you're
doing from now on but please tell me why $$\pi$$ is in the numerator?

 

[toasticles]

i thought it was because original when it is brought up and r is put to the power of a negative it becomes a negative number...? probably wrong what is the right way to do it i always get mixed up with the order of how the terms and numbers go :(

 

[sponsoredwalk]

Lets find out for ourselves what actually happens to $$\pi$$ in this situation.
For convenience let's set 12.75 = a so the last term in the equation becomes

$$\frac{a}{\pi r^2}$$

Now, using the definition of the derivative $$f'(x) \ = \ \lim_{h \to 0} \ \frac{f(x \ + \ h) \ - \ f(x)}{h}$$

calculate the derivative of $$f(r) \ = \ \frac{a}{\pi r^2}$$

I'll start you off

$$f(r \ + \ h) \ - \ f(r)} \ = \ (\frac{a}{\pi (r \ + \ h)^2} \ - \ \frac{a}{\pi r^2})$$

EDIT: Sorry, I messed up by skipping steps, divide through by "h" on both sides at the end

 

[toasticles]

is that the correct answer? I'm sorry i haven't answered your question, i haven't done the limits or the first principles in 2 years :(

also note that the -2 and -3 are meant to be in superscript as powers but i don't know how to work latex like awesomesauce yet

 

[sponsoredwalk]

Everything except the $$\pi$$ is correct, you need to understand why what
you've written is incorrect.

If you just cross multiply the right hand side of the last latex message I wrote & then get it looking pretty, you're almost there. All you have to do then is divide both sides by h
& then take a limit & you'll have an answer.

 

[toasticles]

okay i'll give it a shot

 

[toasticles]

this is what i got for the next step

i can't seem to cross multiply or go any further ... ughhhhh

 

[sponsoredwalk]

$f(r \ + \ h) \ - \ f(r)} \ = \ (\frac{a}{\pi (r \ + \ h)^2} \ - \ \frac{a}{\pi r^2}) \ = \ \frac{r^2}{r^2} \cdot \frac{a}{\pi (r \ + \ h)^2} \ - \ \frac{(r + h)^2}{(r + h)^2} \cdot \frac{a}{\pi r^2} \ = \ \frac{ar^2}{\pi r^2(r + h)^2} \ - \ \frac{a(r + h)^2}{\pi r^2 (r + h)^2}$

$f(r \ + \ h) \ - \ f(r)} \ = \ \frac{ar^2 \ - \ a(r + h)^2}{\pi r^2 (r + h)^2}$

Can you keep going from here?

If you look at the equation you'll see that I multiplied both of the crazy fractions by terms that are, in reality, only equal to 1. Doing this is smart because you're allowed to multiply a number by 1, even if it's written as $$\frac{2}{2}$$ or even $$\frac{(r + h)^2}{(r + h)^2}$$

It's the same thing as "cross multiplying" except this way makes sense as opposed to some magical trick

 

[toasticles]

hmmm okay let me see

 

[toasticles]

yeah? in this case a is 12.75 so pi stays underneath in the denominator? xD?

 

[sponsoredwalk]

lol, I don't think you remember this

www.khanacademy.org has great videos on how to do all of this, you should check them out.

All you have to do here is to expand the a(r + h)² term on top of the fraction, then a lot
of things cancel out by minus signs etc... Then, you're left with a new weird fraction.

Set that up, then as the definition of the derivative implies you divide both sides by "h".

More stuff will cancel out, this happens every time you take a derivative!

All that is left is to take the limit as h-->0 on both sides & you'll have an answer.

I really suggest going to watch the videos at that site &/or getting a good book like
Swokowski's calculus (old edition is 20 cent on amazon & it's amazing!).

 

[toasticles]

so this is the "weird" new fraction?

 

[sponsoredwalk]

Yes, well done! Personally I think you should write everything out fully all the time so that
you know exactly what you're doing & can see the logic to all of it.

I'll just show you what I'm talking about$f(r \ + \ h) \ - \ f(r)} \ = \ \frac{ar^2 \ - \ ar^2 \ - \ 2arh \ - ah^2}{\pi r^2 (r + h)^2}$$f(r \ + \ h) \ - \ f(r)} \ = \ \frac{- \ 2arh \ - ah^2}{\pi r^2 (r + h)^2}$

$\frac{f(r \ + \ h) \ - \ f(r)}{h} \ = \ \frac{- \ 2arh \ - ah^2}{ h \pi r^2 (r + h)^2}$

$\frac{f(r \ + \ h) \ - \ f(r)}{h} \ = \ \frac{- \ 2ar \ - ah}{ \pi r^2 (r + h)^2}$

You see, everything is logically laid out so that you can't make a mistake, try to get things
written out this way until you can do all of this in your head.

All you have to do is to take the limit of both sides now

$\lim_{h \to 0} \ \frac{f(r \ + \ h) \ - \ f(r)}{h} \ = \ \lim_{h \to 0} \ \frac{- \ 2ar \ - ah}{ \pi r^2 (r + h)^2}$

See! On the left hand side you have the definition of the derivative clearly shown!
On the right you need to send h to 0 & look, a term dissappears & the h in the denominator
also dissappears so that you get a clean answer.

$\lim_{h \to 0} \ \frac{f(r \ + \ h) \ - \ f(r)}{h} \ = \ \lim_{h \to 0} \ \frac{- \ 2ar}{ \pi r^2 r^2}$

$\lim_{h \to 0} \ \frac{f(r \ + \ h) \ - \ f(r)}{h} \ = \ \frac{- \ 2ar}{ \pi r^4}$

$\lim_{h \to 0} \ \frac{f(r \ + \ h) \ - \ f(r)}{h} \ = \ \frac{- \ 2a}{ \pi r^3}$

a = 12.75

$\lim_{h \to 0} \ \frac{f(r \ + \ h) \ - \ f(r)}{h} \ = \ \frac{- \ 25.5}{ \pi r^3}$Check out the videos, they'll give you a lot of good explanations for all of this & I do advise
getting that cheap book, it's better than Stewart, Thomas etc...

 

[toasticles]

you sir are a legend and yes actually funnily enough my modern history teacher recommended this site to me when i had spares today, she said it was epic with calculus etc so obviously she wasn't bsing lol xD thanks man i fully see it now (btw i did have it written down in lined book while i was doing it i just really couldn't remember xD)

 

[sponsoredwalk]

lol cool, no problem