Answer:
= -sin(x)
Explanation:
Possible derivation:
d/dx(cos(x))
From the limit definition of the derivative, d/dx(cos(x)) = lim_(h->0) (cos(x + h) - cos(x))/h:
= lim_(h->0) (cos(x + h) - cos(x))/h
Apply the cosine angle addition formula to cos(x + h):
= lim_(h->0) ((cos(x) cos(h) - sin(x) sin(h)) - cos(x))/h
Collect in terms of cos(x) and sin(x):
= lim_(h->0)(cos(x) (cos(h) - 1)/h - sin(x) sin(h)/h)
Multiply numerator and denominator of (cos(h) - 1)/h by the conjugate term cos(h) + 1 and expand the numerator:
= lim_(h->0)(cos(x) (cos^2(h) - 1)/((cos(h) + 1) h) - sin(x) sin(h)/h)
Apply the Pythagorean identity sin^2(h) + cos^2(h) = 1:
= lim_(h->0)(-cos(x) (sin^2(h))/((cos(h) + 1) h) - sin(x) sin(h)/h)
Factor inside the limit:
= lim_(h->0)((-(cos(x) sin(h))/(cos(h) + 1) - sin(x)) sin(h)/h)
The limit of a product is the product of the limits:
= (lim_(h->0)(-(cos(x) sin(h))/(cos(h) + 1) - sin(x))) (lim_(h->0) sin(h)/h)
By continuity, lim_(h->0)(-(cos(x) sin(h))/(cos(h) + 1) - sin(x)) = -(cos(x) sin(0))/(cos(0) + 1) - sin(x) = -sin(x):
= -sin(x) lim_(h->0) sin(h)/h
Apply the common limit lim_(h->0) sin(h)/h = 1:
Answer: = -sin(x)