With this, we have have a few things going on here. First notice the chain rule needed for 3x and then that d/dx sinx = cosx , d/dx cosx = -sin× , d/dx -sinx = - cos and finally d/dx -cosx = sinx. In knowing these derivatives, you know that you need to take the derivative FOUR times to return it back to itself. Doing the 77th derivative makes you do taking the derivative in these 4 time "cycles" 19 times (bc 77/4 = 19.25) which leaves you with taking the derivative just ONCE more after the first 76 times. So the 77th derivative of sinx is cosx. That is not all though. Recognixe that you will also multiply it by 3 77times bc of chain rule, so the 77th derivative of sin (3x) is......: ( 3^77 × cos (3x) )
