find the derivative of cos(ax+h) using the first principle.

Question

hartnn · Answer

do you know the first principle ?
and the function f(x) is just cos (ax) , right ?

anonymous · Answer

I used the principle, and got it until this part:

$$(2\sin(ax+ah+b+ax+b)\div2 - \sin ((ax+ah+b)-(ax+b))\div2)$$

hartnn · Answer

oh, so your function is cos(ax+b)

and you're trying to simplify 
$\cos(a[x+h]+b) - \cos (ax+b)$
which is in the numerator, right ??

mathmale · Answer

DeadMad:  I'd appreciate your defining what you mean by "first principle."  From what you've typed in, and from what hartnn has typed, I take it that you're trying to find the derivative of cos (ax+h) using the "definition of the derivative."  Is that correct or not?

anonymous · Answer

Yep, im using the formula :

$$(f(x+h)-f(x) \div )h$$

anonymous · Answer

it starts out with the numerator hartnn says.

mathmale · Answer

I am assuming that the function you're supposed to differentiate is

$$y=\cos ax$$

hartnn · Answer

its y = cos (ax+b)

mathmale · Answer

You are to form the "difference quotient," which you have partially, but not completely, correct.  The correct form is$$\frac{ f(x+h)-f(x) }{ h }$$

mathmale · Answer

Many thanks, hartnn!  I overlooked that.

mathmale · Answer

If y=f(x)=cos (ax+b), then 

f(x+h)=cos (a[x+h] +b).

Write the difference quotient as

$$\frac{ \cos(a[x+h]+b)-\cos( ax+b)}{h  }$$

mathmale · Answer

Wow.  Looks as tho' you're going to have to simplify this by using the addition rule for the cosine function:$$\cos(a+b)=\cos a \cos b - \sin a \sin b$$

mathmale · Answer

and that you're going to have to do that twice, on each of the cos ... terms.

mathmale · Answer

I need to get off my computer.  Before I do that, however, please fire any questions you may have at this point.

anonymous · Answer

Hmm... I thought we would have to use the cosa-cosb=2sin(a+b)/2 + sin(a-b)/2

anonymous · Answer

I mean * not +.

mathmale · Answer

That may work (although that's not an approach I'd take in this particular case).

You'll need to weigh the relative advantages and disadvantages of each approach:  yours and mine; there may be others as well.

the original function is in cos; what would be the advantage of changing that to sin ((a+b)/2)?  I'm not saying that your idea is not a good one; I just want to know your reasoning.  

Personally, I'd stick with the sum formula for the cosine of (a+b), which I typed in earlier.  But it's your choice.

I'm hitting the sack now; if you choose to respond to my comments here, or if you message me, I'll try to respond again tomorrow morning.  Good night and good luck!