Knowing that derivative of sinx is cos⁡x and derivative of cosx is -senx, prove by induction, that the 2n derivative of sinx = ((- 1) ^ n ) * sin⁡x⁡ and the 2n derivative of cos⁡x = ((- 1) ^ n)* cos⁡x.

Question

Knowing that derivative of sinx is cos⁡x and derivative of cosx  is -senx, prove by induction, that the 2n derivative of sinx = ((- 1) ^ n ) * sin⁡x⁡ and the 2n derivative of cos⁡x  = ((- 1) ^ n)* cos⁡x.

oobleck · Answer

so, check n=1. The 2nd derivative of sinx is -sinx since d/dx(sinx) = cosx d/dx(cosx) = -sinx Now assume it is true for k=2n. What about 2n+2? y^⁽²ⁿ⁾sinx = (-1)^ⁿ sinx y^{⁽²ⁿ⁺¹⁾}sinx = (-1)^ⁿ cosx y^{⁽²ⁿ⁺²⁾}sinx = -(-1)^ⁿ cosx = (-1)^ⁿ⁺¹ sinx QED