Posted by **Fred** on Saturday, April 17, 2010 at 10:28am.

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

Step are needed and the general rule/pattern

- Calculus -
**Reiny**, Saturday, April 17, 2010 at 10:43am
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(-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^{VIII} would be (-4)(-4)e^x(sinx)

etc.

so y^{1000}= (-4)^250(e^x(sinx))

= (4^250)(e^x(sinx))

- Calculus -
**Joey**, Saturday, April 17, 2010 at 10:47am
where did u get the 250 from?

- Calculus -
**Damon**, Saturday, April 17, 2010 at 10:51am
every fourth one is 4 times four before it.

and

1000/4 is 250

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