Derivative of Sin
d/dx sin(x) = cos(x)
The derivative of sin(x) is cos(x), which means that the slope of the function sin(x) at any point x is given by the function cos(x). In other words, the instantaneous rate of change of sin(x) at any point x is equal to cos(x). This derivative can be derived using the limit definition of the derivative:
lim(h -> 0) [sin(x + h) – sin(x)]/h
= lim(h -> 0) [sin(x)cos(h) + cos(x)sin(h) – sin(x)]/h
= lim(h -> 0) [sin(x)(cos(h) – 1)/h + cos(x)sin(h)/h]
= sin(x)lim(h -> 0) (cos(h) – 1)/h + cos(x)lim(h -> 0) sin(h)/h
= sin(x)(0) + cos(x)(1)
= cos(x)
Therefore, the derivative of sin(x) is cos(x).
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