Question

If f(x) = cos(pix), find the value of the 13th derivative of f(x) at x = 2/3.

is there a way to do this without taking the derivative of the derivative of the derivative of the derivative etc 13 times?

Answer 1

No, but it's a pretty easy derivative to take
cos(pix)--->-sin(pix)---->-cos(pix)--->sin(pix)----cos(pix)--->
Notice it comes back to cos(pix) every 4th derivative and repeats the pattern.

Answer 2

yeah so I got to 13th which is -sin(pix) but how would I add in the chain rule?

Answer 3

oops my bad, i messed up slightly

Answer 4

cos(pix)--->-pi*sin(pix)---->-pi^2*cos(pix)--->pi^3*sin(pix)----pi^4cos(pix)--->
fixed.

Answer 5

you gotta keep increasing that pi term outside, 'cause you have to take the derivative of that
pi*x each time, too

Answer 6

so it would be -pi^13*sin(pix) ?

Answer 7

yes!

Answer 8

okay, thank you so much! :D

Answer 9

no problem.