Derivative of sin x
cos x
The derivative of sin x is cos x.
To show this, we can use the definition of the derivative:
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
We can use the identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b) to rewrite the numerator in the above equation:
sin'(x) = lim(h->0) [sin(x)cos(h) + cos(x)sin(h) – sin(x)]/h
Now we can cancel out the sin(x) terms:
sin'(x) = lim(h->0) [cos(x)sin(h)]/h
We can simplify this by using the fact that lim(h->0) sin(h)/h = 1 (which can be proven using L’Hopital’s rule):
sin'(x) = cos(x)
Therefore, the derivative of sin x is cos x.
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