## derivative of sinx

### cosx

The derivative of sinx is cosx.

To derive this result, we can use the definition of derivatives and the trigonometric identities.

We begin with the definition of the derivative:

f'(x) = lim(h -> 0) [f(x+h) – f(x)]/h

Let f(x) = sinx. Then:

f'(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:

sin(x+h) – sin(x) = sin(x)cos(h) + cos(x)sin(h) – sin(x)

= cos(h)sin(x)

We substitute this into the derivative formula:

f'(x) = lim(h -> 0) [cos(h)sin(x)]/h

Using the limit definition of the cosine function:

lim(h -> 0) [cos(h) – 1]/h = 0

We can rewrite the derivative as:

f'(x) = lim(h -> 0) [cos(h)sin(x)]/h

= lim(h -> 0) cos(h) * lim(h -> 0) sin(x)/h

= cos(0) * sin(x)

= cos(x)

Thus:

(f(x))’ = cos(x)

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