Finding Derivatives of Radicals

**alane1994** · October 8th 2012, 12:43 PM

How does one find the derivative of a radical? Ex. $\sqrt{x+6}$

**MarkFL** · October 8th 2012, 12:48 PM

Rewrite the function using a rational exponent, then use the power rule:

$f(x)=\sqrt{x+6}=(x+6)^{\frac{1}{2}}$ and so:

$f'(x)=\frac{1}{2}(x+6)^{-\frac{1}{2}}=\frac{1}{2\sqrt{x+6}}$

**alane1994** · October 8th 2012, 12:57 PM

So, for the second derivative you would use the quotient rule right? And how would that work? You would have to find the derivative of $2\sqrt{x+6}$ which I believe is $-\sqrt{x+6}$ ... am I on the right track?

**Plato** · October 8th 2012, 12:57 PM

Originally Posted by alane1994

How does one find the derivative of a radical? Ex. $\sqrt{x+6}$

Here is a formula that is most useful. If $f(x)$ is a differentiable function and $y=\sqrt{f(x)}$ then $y^{\prime}=\frac{f^{\prime}(x)}{2\sqrt{f(x)}}$ .

Learning that form can be a great time saver.

**alane1994** · October 8th 2012, 01:06 PM

Awesome, that is a cool way to think of it!

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