# Definition of the derivative

• Jan 27th 2014, 03:26 PM
kinhew93
Definition of the derivative
I can't get anywhere with this question, any help would be awsome:

f(x) = srt(3x+1)
find f'(x) using ONLY the definition of the derivative (no power laws etc)

Starting off:

f'(x) = lim [h->0] [f(x+h) - f(x)]/(h)]

= lim [h->0] [sqt(3x+3h+1) - sqt(3x+1)]/h

From here I cannot seem to get rid of the h from the bottom?
• Jan 27th 2014, 03:32 PM
SlipEternal
Re: Definition of the derivative
Multiply top and bottom by the conjugate of $\displaystyle \sqrt{3x + 3h + 1} - \sqrt{3x+1}$. The conjugate is $\displaystyle \sqrt{3x+3h + 1} + \sqrt{3x+1}$. So, you get:

$\displaystyle \lim_{h \to 0} \dfrac{\sqrt{3x+3h+1} - \sqrt{3x+1}}{h}\cdot \dfrac{\sqrt{3x+3h+1} + \sqrt{3x+1}}{\sqrt{3x+3h+1} + \sqrt{3x+1}}$

Now, the numerator simplifies to $\displaystyle 3h$ which is divisible by $\displaystyle h$ allowing you to cancel.

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