Calculus
posted by Fred .
If f(x) = (e^(x))sin(x) what is the 1000th derivative?
Step are needed and the general rule/pattern

y = e^x(sinx)
y^{I} = e^xcosx + e^xsinx
= e^x(cosx + sinx)
y^{II} = e^x(sinx + cosx) + e^x(cosx + sinx)
= e^x(2cosx) = 2e^x(cosx)
y^{III} = 2e^x(sinx) + 2e^x(cosx)
= 2e^x(cosx  sinx)
y^{IV} = 2e^x(sinxcosx) + 2e^x(cosx  sinx)
= 2e^x(2sinx_
= 4e^x(sinx)
Ahhh so it took 4 derivatives to reach 4(what we started with)
so y^{VIII} would be (4)(4)e^x(sinx)
etc.
so y^{1000}= (4)^250(e^x(sinx))
= (4^250)(e^x(sinx)) 
where did u get the 250 from?

every fourth one is 4 times four before it.
and
1000/4 is 250