Absolutely, let's go through this together!
We are given a function, f(x), which is cos(x). We want to find its derivative, which is often represented as f'(x) or frac {d}{dx} f(x).
In calculus, the derivative of a function gives us the rate at which the function is changing at any given point. For the cosine function, or cos(x), the derivative is already well defined and it happens to be -sin(x). There's a straightforward, general rule for this, but you might be interested in seeing why that is.
This can be proven using the limit definition of the derivative:
f'(x) = lim (h -> 0) [cos(x+h) - cos(x)] / h.
We can use the trigonometric addition formula (cos(a+b) = cos(a)cos(b) - sin(a)sin(b)) and apply it to the numerator of our limit:
= lim (h -> 0) [cos(x)cos(h) - sin(x)sin(h) - cos(x)] / h.
Now, let's split this up into two limits:
= cos(x) * lim (h -> 0) [cos(h)-1]/h - sin(x) * lim (h -> 0) sin(h)/h.
When you apply the limits to these fractions, the first one becomes 0 and the second one is 1, due to well-known trigonometric limits. So, you end up with:
= cos(x) * 0 - sin(x) * 1
= -sin(x).
So, we have derived that the derivative of cos(x) is indeed -sin(x).