Derivative of Cos(x) from the first principle 

To find the derivative of the cosine function, cos(x), using the first principles (also known as the limit definition of a derivative), you can start with the definition of the derivative:
f'(x) = lim (h→0) [f(x + h) - f(x)] / h
In this case, f(x) is cos(x). So, you have:
f(x) = cos(x)
Now, apply the limit definition:
f'(x) = lim (h→0) [cos(x + h) - cos(x)] / h
Now, let's simplify this expression. You can use the trigonometric identity for the cosine of the sum of angles:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
In this case, A is x, and B is h:
cos(x + h) = cos(x)cos(h) - sin(x)sin(h)
So, substitute this into the limit definition:
f'(x) = lim (h→0) [cos(x)cos(h) - sin(x)sin(h) - cos(x)] / h
Now, distribute the h into each term in the numerator:
f'(x) = lim (h→0) [cos(x)cos(h)/h - sin(x)sin(h)/h - cos(x)h/h]
As h approaches 0, cos(h)/h approaches 0 and sin(h)/h approaches 1. So, the limit becomes:
f'(x) = cos(x) * 0 - sin(x) * 1 - cos(x) * 0
f'(x) = -sin(x)
Therefore, the derivative of cos(x) with respect to x, using the first principles, is -sin(x).