Year 2015 Differentiation (Easy)

Calculus Level 1

Find the 2015th derivative of f(x) = cos(x) when x = 2015 pi radians.

The answer is 0.

2 solutions

Tony Sprinkle
Jan 4, 2015

The derivatives of $\cos x$ go through a cycle: $f(x) = \cos x$ $f'(x) = -\sin x$ $f''(x) = -\cos x$ $f'''(x) = \sin x$ $f^{(4)}(x) = \cos x$

So, the 4th, 8th, 12th, ... derivatives of $\cos x$ are the function itself. The largest multiple of 4 that's less than or equal to 2015 is 2012, leaving 3 more derivatives. Thus: $f^{(2015)}(x) = f'''(x) = \sin x$

And $\sin (2015\pi) = \boxed{0}$ . (The sine of any integer multiple of $\pi$ is $0$ ).

Can't we simply do this? $\cos (2015^{c})$ will have a constant value, and as the derivative of a constant term is zero, the $2015^{\text{th}}$ derivative will be zero.

Omkar Kulkarni - 6 years, 3 months ago

exactly...cos(2015pi) is a constant. Hence, its nth derivative will be zero

Krishna Ramesh - 6 years, 1 month ago

Avraam Aneleitos
Jan 3, 2015

The nth derivative of cosx is given by cos(x+n*pi/2).With substitution of n=2015 and x=2015pi we get 0.

Year 2015 Differentiation (Easy)

The answer is 0.

2 solutions

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