Derivative of sin x
The derivative of sin(x) can be found using the definition of the derivative or by using the trigonometric identities
The derivative of sin(x) can be found using the definition of the derivative or by using the trigonometric identities.
Using the definition of the derivative, we have:
f'(x) = limit as h approaches 0 of [sin(x+h) – sin(x)] / h
Applying the angle addition formula for sine, we can write:
f'(x) = limit as h approaches 0 of [sin(x)cos(h) + cos(x)sin(h) – sin(x)] / h
Now, dividing each term by h and rearranging, we get:
f'(x) = limit as h approaches 0 of [sin(x)(cos(h) – 1)/h + cos(x)sin(h)/h]
Next, we recognize that as h approaches 0, [cos(h) – 1]/h approaches 0, and sin(h)/h approaches 1. Therefore, we can simplify the expression further:
f'(x) = sin(x)(0) + cos(x)(1)
= cos(x)
Therefore, the derivative of sin(x) is cos(x).
Another way to find the derivative of sin(x) is by using the trigonometric identities. The derivative of sin(x) can be derived from the identity:
sin(x + h) = sin(x)cos(h) + cos(x)sin(h)
Taking the limit as h approaches 0, we get:
lim(h->0) [sin(x + h) – sin(x)] / h = cos(x)
Therefore, the derivative of sin(x) is cos(x).
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