Derivative of sin(x)
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The derivative of sin(x) is cos(x).
To derive this, we can use the definition of the derivative as the limit of the difference quotient:
f'(x) = lim[h->0] [(f(x+h) – f(x))/h]
We substitute sin(x) for f(x) in this equation:
f'(x) = lim[h->0] [(sin(x+h) – sin(x))/h]
Then we can use the trigonometric identity that sin(a+b) = sin(a)cos(b) + cos(a)sin(b) to rewrite sin(x+h) as a combination of sin(x) and cos(x):
f'(x) = lim[h->0] [(sin(x)cos(h) + cos(x)sin(h) – sin(x))/h]
We simplify this expression by factoring out sin(x) from the first two terms and canceling the h from the denominator:
f'(x) = lim[h->0] [(sin(x)(cos(h)-1))/h + cos(x)sin(h)/h]
Now we can take the limit as h approaches 0. The first term goes to 0 because cos(h)-1 approaches 0 as h approaches 0. The second term approaches cos(x) because sin(h)/h approaches 1 as h approaches 0. Therefore:
f'(x) = cos(x)
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