Thread: differentiation from first principals

1. differentiation from first principals

Hi Math help forum!

I'm finding it difficult to differentiation from first principals and wanted to know if there is a simple method I can follow. Could you please give an example aswell thanks.

regards

dadon

2. "from first principles" would be using the (limit) definition?

3. Thanks for the reply

Yes any method as long as it easy to follow and you get the right answer!

4. Well there is no standard 'way' of doing them, since evaluating limits can go very different, depending on what limit it is. For what kind of functions do you have to be able to get its derivative through the definition?

5. Just so I'm clear, is this what you are talking about?

$\displaystyle f'(x)=\lim_{\Delta{x}\rightarrow{0}}\frac{f({x}{+} \Delta{x})-f(x)}{\Delta{x}}$?

6. I realized that you used
Code:
\rightarrow
While you can have used,
Code:
\to

7. yeah jameson when the limit turns zero.

8. Originally Posted by dadon
yeah jameson when the limit turns zero.
That's what I wrote. Hence the $\displaystyle \Delta{x}\to{0}$. What seems to be the problem? These problems usually require some basic algebra manipulation. Try finding the derivatives of $\displaystyle f(x)=x^2$ and $\displaystyle f(x)=\frac{1}{x}$ using the limit definition. Or is there a specific one we can help you with?

9. PerfectHacker:

Thanks. A small time saver.

10. Originally Posted by Jameson
PerfectHacker:

Thanks. A small time saver.
I also made the same thing, then I was curious to see what Code TD! used and a saw he used a simpler one used it ever since.

11. re:

So say I had to find from first principals $\displaystyle \frac{dy}{dx}$ of the following:

$\displaystyle y = 16x + \frac{1}{x^2}$

cheers guys

12. Set up your limit.

$\displaystyle \frac{dy}{dx}=\lim_{\Delta{x}\to{0}}\frac{16(x{+}{ \Delta}x)+\frac{1}{(x{+}{\Delta}{x})^2}-16x-\frac{1}{x^2}}{\Delta{x}}$

Is this where you're having trouble?

13. thanks for the reply.

yes that is what i needed help on.

14. Originally Posted by dadon
thanks for the reply.

yes that is what i needed help on.
You do not need to spend much time and master that limit. Because soon you will learn rules which would make it easier.