1. ## using maclaurin series that we already know

so, i have two really similar questions here:

1) using the maclaurin series for $\displaystyle e^x$, show that $\displaystyle (e^x)' = e^x$

and 2) using the maclaurin series for $\displaystyle sin(x)$, show that $\displaystyle (sin(x))'= cos(x)$

i know what these are, but, i have no idea how to show what they're asking

any help would be greatly appreciated!

2. Originally Posted by buttonbear
so, i have two really similar questions here:

1) using the maclaurin series for $\displaystyle e^x$, show that $\displaystyle (e^x)' = e^x$

and 2) using the maclaurin series for $\displaystyle sin(x)$, show that $\displaystyle (sin(x))'= cos(x)$

i know what these are, but, i have no idea how to show what they're asking

any help would be greatly appreciated!
$\displaystyle e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ... + \frac{x^n}{n!} + ...$

take the derivative of both sides ... what do you see?

now do the same for the maclaurin series for $\displaystyle \sin{x}$

3. wait.. how do you take the derivative of factorial?

4. Originally Posted by buttonbear
wait.. how do you take the derivative of factorial?
you don't ... it's a constant multiple of a term of the function.

for example, the derivative of $\displaystyle \frac{x^3}{3!}$ is $\displaystyle \frac{3x^2}{3!} = \frac{x^2}{2!}$

5. oh..obviously
forgive me, i'm sleep deprived (among other things)
thank you!