Find the Derivative of cos x from First Principle

Answer: According to the first principle, derivative of a function f(x) is given by 

f'(x) = lim_h→0 [f(x+h)-f(x)] /h

So, if we consider f(x)=cos x, then its derivative is given by 

f'(x) = lim_h→0 [cos (x+h) – cos(x)] /h = lim_h→0 (cos x. cos h – sin x. sin h – cos x) /h

So, f'(x) = lim_h→0[ {-cos x (1-cos h)} /h – {sin x. sin h} /h]

So, f'(x) = -cos x lim_h→0 (1-cos h)/h – sin x lim_h→0 sin h /h

Using the formula and identities of limits, we know, lim_x→0 sin x /x = 1 and lim_x→0  (1-cos x) /x = 0

Using these identities, we can write, 

f'(x) = -cos x (0) – sin x (1) = 0 – sin x = 0

So, using the first principle, we found that the derivative of cos x is -sin x.