Question

find the nth derivative of cos^6x​

Answer 1

Step-by-step explanation:

f(x)=cos6x

f'(x)=-6cos5(x) sin(x)=-3cos4(x)sin(2x)

f''(x)=-6(-5cos4(x)sin2(x)+cos6(x) )=6cos4(x)(2-3cos(2x))

f'''(x)=-120cos3(x) sin3(x)+60cos5(x) sin(x)+36cos5(x) sin(x)

         =12cos3(x)(-1 + 9cos(2x))sin(x)

 

f(iv)(x)=6cos2(x)(11-22cos(2x)+27cos(4x))

f(v)(x)=-3(5sin(2x)+64sin(4x)+81sin(6x))

f(vi)(x)=-6(5cos(2x)+128cos(4x)+243cos(6x))

f(7)(x)=12(5sin(2x)+256sin(4x)+729sin(6x))

f(8)(x)=24(5cos(2x)+512cos(4x)+2187cos(6x) )

 

f(2n-1)(x)=(-1)n+13*2n-2(5sin(2x)+2n+3sin(4x)+3n+1sin(6x)), n=3,4,..

f(2n)(x)=(-1)n3*2n-1(5cos(2x)+2n+4cos(4x)+3ncos(6x)),n=3,4,...