derv sin(x)
cos(x)
The question appears to be incomplete since derv sin(x) is not a complete statement or question. It is possible that the intended question is What is the derivative of sin(x)?
The derivative of sin(x) is cos(x). This means that the slope of the curve of sin(x) at any point x is equal to the value of cos(x) at that point x.
We can prove this using the definition of derivative:
Let f(x) = sin(x). Then the derivative of f(x) is given by:
f'(x) = lim(h->0) (f(x + h) – f(x))/h
Substituting sin(x) for f(x), we get:
f'(x) = lim(h->0) (sin(x + h) – sin(x))/h
We can use the trigonometric identity for the sine of sum of two angles to rewrite the numerator as:
f'(x) = lim(h->0) (2cos((x + h + x)/2)sin(h/2))/h
Taking the limit as h->0, the term sin(h/2)/h approaches 1, and we obtain:
f'(x) = lim(h->0) 2cos((2x + h)/2)
Simplifying this expression, we have:
f'(x) = 2cos(x)
Therefore, the derivative of sin(x) is cos(x).
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