Find the derivative of cosine of a number by first principle of differention ?

To find the derivative of cosine using the first principle of differentiation, we start with the definition of the derivative:

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

In this case, our function is f(x) = cos(x). So, we substitute this into the definition:

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

Next, we can use the trigonometric identity cos(a+b) = cos(a)cos(b) - sin(a)sin(b) to simplify the expression:

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

Now, we can factor out cos(x) from the numerator:

f'(x) = lim(h->0) [cos(x)(cos(h) - 1) - sin(x)sin(h)] / h

Next, we can use the trigonometric identity sin(a)/a = 1 to simplify the expression further:

f'(x) = lim(h->0) [cos(x)(cos(h) - 1) - sin(x)sin(h)] / h * sin(h)/sin(h)

This allows us to cancel out the sin(h) terms:

f'(x) = lim(h->0) [cos(x)(cos(h) - 1) - sin(x)sin(h)] / h * sin(h)/sin(h)
= lim(h->0) [cos(x)(cos(h) - 1)/h - sin(x)sin(h)/h]

Finally, we take the limit as h approaches 0:

f'(x) = cos(x)(lim(h->0) (cos(h) - 1)/h) - sin(x)(lim(h->0) sin(h)/h)

The limit of (cos(h) - 1)/h as h approaches 0 is 0, and the limit of sin(h)/h as h approaches 0 is 1. Therefore, the derivative of cosine is:

f'(x) = -sin(x)