Need help with a calculus problem

Question

Find d⁸³ / dx⁸³ cos (-0.5x).

Answer 1

For f(x) = cos(x) then

f '(x) = -sin(x)

f ''(x) = -cos(x)

f '''(x) = sin(x)

f ''''(x) = cos(x)

So d⁴/dx⁴ = cos(x)

d⁸/dx⁸ = cos(x)

d¹²/dx¹² = cos(x)

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d⁸⁰/dx⁸⁰ = cos(x)

d⁸¹/dx⁸¹ = -sin(x)

d⁸²/dx⁸² = -cos(x)

d⁸³/dx⁸³ = sin(x)

But in this case, the chain rule is in play. So each time we take a derivative we must multiply by the derivative of what's inside. That is -0.5 so:

d⁸³/dx⁸³ = (-0.5)⁸³sin(-0.5x)