How do you compute the 200th derivative of #f(x)=sin(2x)#?

1 Answer
Ananda Dasgupta
May 21, 2018

#2^200 sin(2x)#

Explanation:

#f(x) = sin(2x) implies#
#d/dx f(x) = 2 cos(2x) implies#
#d^2/dx^2 f(x) = -2^2 sin(2x) implies#
#d^3/dx^3 f(x) = -2^3 cos(2x) implies#
#d^4/dx^4 f(x) = 2^4 sin(2x) implies#

This means that differentiating #f(x)# 4 times results in the same function, with a multiplying factor of 4.

Differentiating 200 times is the same as repeating the above 50 times, and so

#d^200/dx^200 f(x) = (2^4 )^50sin(2x) = 2^200 sin(2x)#

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