finding the trignometric derivative

catalle

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Joined
Oct 5, 2008
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9
so I did some steps for this problem and got stuck halfway down, please help

find dy/dx
y=1+x-cos(x)
= [1-cos(x)] + x
= sin(x) + x

[sin (x+h) + (x+h) - sin (x) - x]/h --- definition of derivative

[sin(x)*cos(h) + cos(x)*sin(h) + x + h - sin(x)-x] / h --- angle sum rule?

[sin(x)*cos(h) + cos(x)*sin(h) + h - sin(x)] / h --- eliminate Xs

do I eliminate the Hs now as well? but then there wouldn't be an h on the denominator
do I plug in 0 for h now or what?
 
1cos(x)sin(x)\displaystyle 1-cos(x)\neq sin(x)

Unless you have a typo.

Do you HAVE to use the def. of a derivative approach?.
 
You are correct about using the addition rule for cos.

limh01+(x+h)cos(x+h)1x+cos(x)h\displaystyle \lim_{h\to 0}\frac{1+(x+h)-cos(x+h)-1-x+cos(x)}{h}

limh0hcos(x)cos(h)+sin(x)sin(h))+cos(x)h\displaystyle \lim_{h\to 0}\frac{h-cos(x)cos(h)+sin(x)sin(h))+cos(x)}{h}

Factor out a cos(x) and a sin(x). Then, notice the famous limits cos(h)1h=0,   sin(x)x=1\displaystyle \frac{cos(h)-1}{h}=0, \;\ \frac{sin(x)}{x}=1

limh01cos(x)(cos(h)1h)+sin(x)(sin(x)x)\displaystyle \lim_{h\to 0}1-cos(x)\left(\frac{cos(h)-1}{h}\right)+sin(x)\left(\frac{sin(x)}{x}\right)
 
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