Use primary trig identities to determine the derivative of y= cot x

Question

a) Use primary trig identities to determine the derivative of y= cot x.

b) then use the result from a) to determine f"(x), where f(x)= cot 4x

Answer 1

y = cotx = cosx/sinx

y' = [sinx(cosx)' - cosx(sinx)')]/sin^2x = -sinx/sinx - cot^2(x) = -(cot^2(x) +1) =-csc^2(x)

f(x) =cot4x

f'x) = -4csc^24x

f"(x) = -8csc4x(-csc4xcot4x)(4) = 32csc^2xcotx =

32csc^24xcos4x/sin4x =

32csc^2xcos4xcsc4x =

32csc^3(4x)cos(4x)