Question

y=sin^4x then find its n th discrivation

Answer 1

Answer:

Step-by-step explanation:

Here, f(x) = sin 2x sin 4x 

sin(a)sin(b) = (1/2) ( cos(a-b) - cos(a+b)) 

sin 2x sin 4x = (1/2) ( cos(-2x) - cos(6x) ) = (1/2) cos(2x) - (1/2) cos(6x) 

f'(x) = (1/2) (-2 sin 2x) - (1/2) (-6 sin 6x) 

f'(x) = -sin 2x + 3 sin 6x 

f''(x) = - 2 cos 2x + 18 cos 6x 

f'''(x) = 4 sin 2x - 108 sin 6x 

f^4(x) = 8 cos 2x - 648 cos 6x 

f^5(x) = -16 sin 2x + 3,888 sin 6x 

f^6(x) = -32 cos 2x + 23,328 cos 6x 

f^7(x) = 64 sin 2x - 139,968 sin 6x 

Now, the odd derivatives have the sum of two sine terms 

and, the even derivatives have the sum of two cosine terms 

So, nth derivative: 

If n is even: 

(-1)^(n-1) [ 2^(n-1) cos 2x - 3*6^(n-1) cos 6x ] 

If n is odd: 

f^n(x) = [-2^(n-1) sin 2x + 3*6^(n-1) sin 6x] 

or 

f^n(x) = [2^(n-1) sin 2x - 3*6^(n-1) sin 6x]

hope it helps...................................................................

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