Question

Find nth derivative of y = Sin(x) Cos^3(x)

(a) (1/4) 2^nSin(2x+nπ/2) + (1/8) 4^nSin(4x+nπ/2)

(b) (1/4) 2^nCos(2x+nπ/2) + (1/8) 4^nSin(4x+nπ/2)

(c) (1/4) 2^nSin(2x+nπ/2) + (1/8) 4^nCos(4x+nπ/2)

(d) (1/4) 2^nCos(2x+nπ/2) + (1/8) 4^nCos(4x+nπ/2)

The question was asked by my school teacher while I was bunking the class.

Origin of the question is The nth Derivative of Some Elementary Functions topic in chapter Differential Calculus of Engineering Mathematics

Answer 1

The correct answer is (a) (1/4) 2^nSin(2x+nπ/2) + (1/8) 4^nSin(4x+nπ/2)

The explanation is: y = Sin(x) Cos^3(x)

y = (1/2)[2Sin(x) Cos(x)] Cos^2(x)

y = (1/4)Sin(2x)(Cos(2x)+1)

y = (1/4)[Sin(2x)Cos(2x)+Sin(2x)]

y = (1/8)Sin(4x) + (1/4)Sin(2x)

Hence nth derivative of y is

Yn = (1/4) 2^nSin(2x+nπ/2) + (1/8) 4^nSin(4x+nπ/2)