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Def. of derivative and cosx=sin(Pi/2-x) to prove y'=-sinx

  1. Jan 5, 2016 #1
    A lot of web pages/books show how to use cosx=sin(Pi/2-x) and the chain rule to prove that the derivative of
    cosx=-sinx. My question is how to use this identity and the defintion of the derivative to prove the same thing.
    Or whether it is at all possible. Seeing that i get dy/dx=(cos(x+h)-cosx)/h = (sin(Pi/2-x-h)-sin(Pi/2-x))/h =
    (sin(Pi/2-x)cosh-cos(pi/2-x)sinh-sin(pi/2-x))/h. The first term here is the problem, namely sin(Pi/2-x)cosh/h, since this needs to be equal to sin(Pi/2-x) for this to work. Is there another way around?

    Another way of thinking is to say that cosh goes to 1 as h goes to 0, but using the same reasoning with sinh leaves me with dy/dx=0.
     
  2. jcsd
  3. Jan 5, 2016 #2

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    Just continue from there. The way you proceed is to express ##\cos h## and ##\sin h## in terms of their respective Taylor expansion. Let's see what you get after doing this. For example, substituting the expansion for ##\cos h## to the first term leads to
    $$
    \sin(\pi/2-x)\cos h = \sin(\pi/2-x)\left(1-\frac{h^2}{2!}+\frac{h^4}{4!} - \ldots \right)
    $$
    Now substract the 3rd term ##\sin(\pi/2-x)## from the RHS of the above equation.
     
  4. Jan 5, 2016 #3

    Mark44

    Staff: Mentor

    Rather than using the identity you're trying to use, just continue from the above.
    ##\frac{\cos(x + h) - \cos(x)}{h} = \frac{\cos(x)\cos(h) - \sin(x)\sin(h) - \cos(x)}{h} = \frac{\cos(x)(\cos(h) - 1)}{h} - \frac{\sin(x) \sin(h)}{h}##.
    To get the derivative of cos(x), take the limits of the two expressions above.
     
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