For the function f(x) = sin (-x), as the sine function is odd, f(x) = -sin x. If F(n) represents the nth derivative,

F(0) = -sin x

F(1) = -cos x

F(2) = sin x

F(3) = cos x

F(4) = -sin x

F(5) = -cos x

It is seen...

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For the function f(x) = sin (-x), as the sine function is odd, f(x) = -sin x. If F(n) represents the nth derivative,

F(0) = -sin x

F(1) = -cos x

F(2) = sin x

F(3) = cos x

F(4) = -sin x

F(5) = -cos x

It is seen that the derivative repeats after the function has been differentiated 4 times.

f(99) = f(3 + 4*24) = f(3) = cos x

**The 99th derivative of f(x) = sin(-x) is cos x**