SOLUTION: Use the definition of the derivation to show that (d/dx)(cos x)= -sin x.
You can use the following 3 facts:
lim h->0 (sin h)/h =1
lim h->0 (cos h -1) / h = 0
cos(θ + )
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Question 486705: Use the definition of the derivation to show that (d/dx)(cos x)= -sin x.
You can use the following 3 facts:
lim h->0 (sin h)/h =1
lim h->0 (cos h -1) / h = 0
cos(θ + )= cosθcos - sinθsin
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Use the definition of derivative using limits:
) = \lim_{\Delta x \to 0} \frac{\cos(x+\Delta x) - \cos(x)}{\Delta x})
 - \sin x \sin (\Delta x) - \cos x}{\Delta x} = \lim_{\Delta x \to 0} \frac{\cos x (\cos (\Delta x) - 1)}{\Delta x} - \frac{\sin x \sin(\Delta x)}{\Delta x})
Applying our known facts, we can treat cos x and sin x as constants and say that the first limit collapses to zero, and the second limit collapses to -sin x. Hence,
 = -\sin x)
We could also use L'Hopital's rule to evaluate the limits, but that assumes we already know the derivative of cos x, so we shouldn't use it right now.
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