The Derivative of the Sine Function: A Step-by-Step Guide to Calculating and Understanding the Derivative of sin(x)

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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