derivative of sin x
The derivative of the sine function, denoted as `sin(x)`, can be calculated using the rules of differentiation
The derivative of the sine function, denoted as `sin(x)`, can be calculated using the rules of differentiation. The derivative of sin(x) is cos(x), where `cos(x)` represents the cosine function.
To derive the derivative of sin(x), we can use the definition of the derivative:
d(sin(x))/dx = lim(h->0) [sin(x+h) – sin(x)] / h
Let’s simplify this expression step by step:
= 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
Now, let’s evaluate the limits as h approaches 0:
The first term, lim(h->0) (cos(h) – 1)/h, represents the derivative of the cosine function at x=0. By definition, this derivative equals 0.
The second term, lim(h->0) sin(h)/h, can be evaluated using L’Hôpital’s rule. L’Hôpital’s rule states that if we have an indeterminate form 0/0 when taking the limit, we can differentiate the numerator and denominator separately and then evaluate the limit again. Applying L’Hôpital’s rule, we get:
lim(h->0) sin(h)/h = lim(h->0) cos(h) = cos(0) = 1
Therefore, the derivative of sin(x) is:
d(sin(x))/dx = 0 + cos(x) = cos(x)
In conclusion, the derivative of the sine function sin(x) is the cosine function cos(x).
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