Derivative of Sin
Cos
The derivative of sin x is cos x.
To prove this, we can start with the definition of the derivative:
f'(x) = lim(h->0)[f(x+h) – f(x)] / h
Let f(x) = sin x. Then, using the angle addition formula for sine, we have:
f(x+h) = sin(x+h) = sin x cos h + cos x sin h
Substituting these expressions into the derivative formula and simplifying, we get:
f'(x) = lim(h->0)[sin x (cos h – 1) + cos x sin h] / h
Using the limit identity lim(h->0)[1-cos h]/h = 0 and lim(h->0)sin h/h =1, we can rewrite the above equation as:
f'(x) = cos x lim(h->0)sin h/h + sin x lim(h->0)[cos h – 1]/h
Since lim(h->0)[cos h – 1]/h = lim(h->0)cos h/h – lim(h->0)1/h = 0 – 0 = 0, we are left with:
f'(x) = cos x
Therefore, the derivative of sin x is cos x.
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