# Thread: Differentiation from 1st principles

1. ## Differentiation from 1st principles

Can anyone see where I'm going wrong here...

So differentiate from 1st principles.

f(x) = sqrt(x)+1

f'(x) = (sqrt(x+h)+1)-(sqrt(x)+1) / h

f'(x) = x+h+1-x-1 / h(sqrt(x+h)+1)+(sqrt(x)+1)

So cancelling terms i get,

f'(x) = 1 / (sqrt(x+h)+1)+(sqrt(x)+1)

f'(x) = 1 / 2sqrt(x) + 2

This is where I'm confused, shouldn't it be

f'(x) = 1 / 2sqrt(x)

2. You did not expand nor rationalize correctly.

\begin{aligned}
\sqrt{x+h} + 1 - (\sqrt{x}+1) &= \sqrt{x+h} + 1 - \sqrt{x} - 1\\
&= \sqrt{x+h} - \sqrt{x}
\end{aligned}

A fit conjugate would be $\sqrt{x+h} + \sqrt{x}$. By your way, you should've got:

$((\sqrt{x+h}+1)-(\sqrt{x}+1))(\sqrt{x+h}+1)+(\sqrt{x}+1)) =(\sqrt{x+h}+1)^2 - (\sqrt{x}+1)^2 = \ldots$

3. Ah i see.. I didn't get rid of the ones... thats a stupid mistake..

Thanks alot