differentiating binomials (1 Viewer)

[Dumbarse]

hey how do u differentiate/integrate a binomial??

and those binomial questions when u have to prove such and such equals such and such, sometimes u have to differentiate or integrate to get the answer, is there a way of knowing when to do this??

 

[Lazarus]

I hope I interpreted your question(s) correctly.


d/dx[ (1 + ax)ⁿ ]

= na(1 + ax)^{n - 1}

The expansion of the derivative will be equal to the term-by-term derivative of the original expansion.

A similar approach can be taken when integrating binomials.


To know whether you need to integrate or differentiate, compare the expansion to the elementary binomial expansions.

A common one you should know is the case when n = -1:

(1 - x)^-1 = 1 + x + x² + x³ + ...

If the series in your question is:

1 + 2x + 3x² + 4x³ + ...

It's fairly obvious that it will be the derivative of (1 - x)^-1.

Similarly, if the series in your question is:

x + x²/2 + x³/3 + ...

It's fairly obvious that it will be the integral of (1 - x)^-1.

I think most questions revolve around different forms of those series. Although I'm not 100% sure. Hope it helped.

 

[spice girl]

My advice would be: If in doubt, try it anyway.

Tricks like integrating/differentiating are like cards - u don't always need to use them. It's good to have them in your hand anyway.

When a question involves integrating/differentiating, when you use the trick to goto the next step, it's usually f***ing obvious it's the right way. If you think you're getting nowhere after 2 steps, forget it.

Try 2001 HSC4u Q8. It's not quite binomials, but if you recognise that something is an integral/differential of something else, use the trick. Not sure? Look harder.

I'm not being specific, i know, but it's the general idea that's important.