Question
Knowing that derivative of sinx is cosx and derivative of cosx is -senx, prove by induction, that the 2n derivative of sinx = ((- 1) ^ n ) * sinx and the 2n derivative of cosx = ((- 1) ^ n)* cosx.
Answers
oobleck
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<sup><sup>(2n)</sup></sup>sinx = (-1)<sup><sup>n</sup></sup> sinx
y<sup><sup>(2n+1)</sup></sup>sinx = (-1)<sup><sup>n</sup></sup> cosx
y<sup><sup>(2n+2)</sup></sup>sinx = -(-1)<sup><sup>n</sup></sup> cosx = (-1)<sup><sup>n+1</sup></sup> sinx
QED
d/dx(sinx) = cosx
d/dx(cosx) = -sinx
Now assume it is true for k=2n. What about 2n+2?
y<sup><sup>(2n)</sup></sup>sinx = (-1)<sup><sup>n</sup></sup> sinx
y<sup><sup>(2n+1)</sup></sup>sinx = (-1)<sup><sup>n</sup></sup> cosx
y<sup><sup>(2n+2)</sup></sup>sinx = -(-1)<sup><sup>n</sup></sup> cosx = (-1)<sup><sup>n+1</sup></sup> sinx
QED