Derivative of sin(x)
cos(x)
The derivative of sin(x) is cos(x).
To understand this, we can use the definition of the derivative, which is the limit of the difference quotient:
f'(x) = lim (h -> 0) [f(x+h) – f(x)] / h
For sin(x), this becomes:
sin'(x) = lim (h -> 0) [sin(x+h) – sin(x)] / h
Using the angle addition formula for sine, we can simplify this to:
sin'(x) = lim (h -> 0) [2cos(x+h/2) sin(h/2)] / h
As h approaches zero, sin(h/2) / (h/2) approaches 1, so we can substitute that in:
sin'(x) = lim (h -> 0) [2cos(x+h/2)] = 2cos(x)
Therefore, the derivative of sin(x) is cos(x).
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