Asked by Fred

If f(x) = (e^(x))sin(x) what is the 1000th derivative?

Step are needed and the general rule/pattern
Answered by Reiny
y = e^x(sinx)
y<sup>I</sup> = e^xcosx + e^xsinx
= e^x(cosx + sinx)
y<sup>II</sup> = e^x(-sinx + cosx) + e^x(cosx + sinx)
= e^x(2cosx) = 2e^x(cosx)
y<sup>III</sup> = 2e^x(-sinx) + 2e^x(cosx)
= 2e^x(cosx - sinx)

y<sup>IV</sup> = 2e^x(-sinx-cosx) + 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<sup>VIII</sup> would be (-4)(-4)e^x(sinx)
etc.
so y<sup>1000</sup>= (-4)^250(e^x(sinx))
= (4^250)(e^x(sinx))
Answered by Joey
where did u get the 250 from?
Answered by Damon
every fourth one is 4 times four before it.
and
1000/4 is 250
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