[tex] \frac{dy}{dx} cos x = -sin x[/tex]
[tex] \frac{dy}{dx} sin x = cos x[/tex]
given that y=cos x and finding its 50th derivative you'll have:
1st:
dy/dx cos x = -sin x
2nd:
y" -sin x = -cos x
3rd:
y'" -cos x = sin x
4th:
y"" sin x = cos x
5th:
y""' cos x = -sin x
notice that the answers would just be in a cycle making certain differentiations of the following equal:
1,5,9,13,17,21,25,29,33,37,41,45,49
2,6,10,14,18,22,26,30,34,38,42,46,50
3,7,11,15,19,23,27,31,35,39,43,47
4,8,12,16,20,24,28,32,36,40,44,47
since the 50th differentiation lies with 2 then, the 2nd derivative is having the same value as the 50th derivative and you'll have it as:
[tex] \frac{ d^{50}y }{d x^{50} } cos x = -cos x [/tex]
