find the first 5 terms of its Maclaurin series: http://puu.sh/ggAqw/2547ca15c1.png
is there any other way other than mechanically find its derivative one after another?

Question

find the first 5 terms of its Maclaurin series: http://puu.sh/ggAqw/2547ca15c1.png
is there any other way other than mechanically find its derivative one after another?

tkhunny · Answer

Absolutely.  This is a good one for the extended Binomial Theorem.

Work on this part: $\left(8+3x^{2}\right)^{-1/3}$

It may be easier first to work with the antiderivative.  $(1/4)\left(8 + 3x^{2}\right)^{2/3}$ and then differentiate term-by-term.  This often works nicely.

anonymous · Answer

@tkhunny why do u want to find its integral?

anonymous · Answer

@robtobey

anonymous · Answer

@hartnn @iambatman @Hero

anonymous · Answer

@myininaya @ikram002p

anonymous · Answer

$$f(x)=f(0)+\frac{ f'(0)x }{ 1! }+\frac{ f''(0)x ^{2} }{ 2! }+...+\frac{ f^{(n)}x^{n} }{ n! }$$

anonymous · Answer

@satellite73 but why tk used integral??? what is that for

anonymous · Answer

@VeritasVosLiberabit

tkhunny · Answer

I gave two choices.  One did not require the integral.

Both choices require the Extended Binomial Theorem.

The integral might be convenient for two reasons...

1) It's relatively easy to find, and
2) The Extended Binomial Theorem may be more obvious to implement.

It's up to you to choose.  I'm just giving you choices.

tkhunny · Answer

Look at "Proposition 22B" on Page 4. https://rutherglen.science.mq.edu.au/wchen/lnfycfolder/fyc22.pdf