Nth derivative of Sin^(4)x
In this quick tutorial you will learn that how to find the nth derivative of a trigonometric function like Sin(x) or Sin^4(x) by successive differentiation method. So observe each and every step very carefully and meanwhile if you have any doubt related to the solution at any step please leave us a comment we assure you that you will hear from us as soon as possible. So let’s now…..
Given: y = sin4x
Multiply and divide sin4x by 4 we get
y = (2sin2x)24
Or y = (1-cos2x)24 Since (2sin2x)2=(1-cos2x)2
Now expanding (1-cos2x)2 we get the following result
i.e y = [(1+cos22x-2cos2x)4]
Again, multiply and divide the term (cos2x)2 with 4 we get the following result.
i.e., y = [(1+4cos22x4-2.cos2x)4]
Or y = [14+116+116(1+cos4x)2-12cos2x]
Or y = [14+116+116(1+cos24x+2cos4x)-12cos2x]
Again Multiply & Divide `cos^(2)4x by 2 we get a result which is differentiable easily.
Or y = [14+116+116(1+2cos24x2+2cos4x)-12cos2x]
Or y = [14+116+116(1+1+cos8x2+2cos4x)-12cos2x]
Or y = [14+116+1161+132+cos8x32+cos4x8-12cos2x]
Now since nth derivative of sin(ax+b) is:
= [an.sin(ax+b+n.π2]
So, now writing nth derivative of each term we get.
= 1328n.sin(8x+n.π2) - 184n.sin(4x+n.π2) -
`\frac{1}{2}{.2^n}.\cos (2x + n.\frac{\pi }{2})`
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