Calculus Help: Diff. y= (int. a=1 b=sinx of) t^3 dt & y=log base x (2x)

[Yapper]

I am studying for a Calculus test and I need some help withsome concepts...


1) The problem is find the derivatie of y= (integral a=1 b=sinx of) t^3 dt. I went into my book and found a formula for it. The derivative of (intergral a=constant b=x of) f(t) dt = f(x). So I got (sinx)^3 is this right? If so can someone explain this concept and provide a proof for it? My textbook isn't good at explaining things...

2)The problem is to differentiate y=log base x (2x). So I chamged it to ln2x/lnx from there can i just use divison rule? u'v -uv'/v^2? in that case would it be (lnx/x-ln2x/x)/(lnx)^2?


Thanks in advance for your help

 

[vincentchan]

1: is your derivative repect to x or t? this make a huge different
2: you are perfectly correct...

 

[Yapper]

y= (integral a=1 b=sinx of) t^3 dt, so the differential unit is t, since it is dt, I am guessing, that's all the information i was given

 

[dextercioby]

y= (integral a=1 b=sinx of) t^3 dt, so the differential unit is t, since it is dt, I am guessing, that's all the information i was given

Then it's wrong.Compute the antiderivative of the integrand and apply the fundamental formula of Leibniz & Newton.

Daniel.

 

[Yapper]

Then it's wrong.Compute the antiderivative of the integrand and apply the fundamental formula of Leibniz & Newton.

Daniel.

No clue what you just said...

 

[Yapper]

How do I use LaTex to make it clearer?

 

[vincentchan]

OK, your integrand is t^3 and the limit is from 1 to sinx... right?
Here is how u do the derivative of this monster...
if it is repect to t.. do the integral and substitude 1 and sinx in... and do the derivative
if it is repect to x.. do the integral and substitude 1 and sinx in... and do the derivative

basically, what you going to do is do the integral and substitude 1 and sinx in... and do the derivative, OK?

but if the derivative is repect to t, the result is trivial...

 

[Yapper]

I don't understand... I do the integral and I get t^4/4 substitutre for sinx and do the derivative I get sinx^3cosx is that right?

 

[dextercioby]

How do I use LaTex to make it clearer?

There's a compiler that will "translate" all code lines u type in the window in which u write the message.Use the function "preview post" to check formulas for typos.

The result is trivial.It can be shown that
\frac{d}{dx}{\int_{const.}^{f(x)} u(t) dt =u(f(x))f'(x)

I still reccomend to you to do the integration and then the differentiation and confront with the result my formula gives you.

Daniel.

 

[vincentchan]

\frac{d}{dx}{\int_{const.}^{f(x)} u(t) dt =u(f(x)) \frac{d}{dx}f(x)

 

[Yapper]

Oh ok... my books says g(x) = (intergral from a constant to x of) f(t) dt, then g'(x) = f(x)

 

[Yapper]

Thanks for all the help! and is there a website or a help file that has the code?

 

[cepheid]

Thanks for all the help! and is there a website or a help file that has the code for the tex imputs?

Our own enormous thread about it here:

https://www.physicsforums.com/showthread.php?t=8997

The first post in this thread has links to external sources about LaTeX

 

[dextercioby]

Thanks for all the help! and is there a website or a help file that has the code for the tex imputs?

Yes,PF is one of them.Check out the "sticky" in the General Physics forum called "Intriducing LaTex typesetting".U'll find amog those posts a link to a ".pdf"file with the code.
Quicker version.Click (left) on one of the formulas in this (and any other one) thread and u'll be opened (if u don't have a pop-up blocker) a new window with the code for that specific formula and a link to the ".pdf" file with the code.

And it's "inputs"...

Daniel.