Need help finding the derivative.

**Darkhrse99** · June 21st 2009, 09:37 PM

I need to find the derivative of

$x-2 \sqrt{x-1}$

I knew to use the power rule, but I was coming up with
$1/2(x-1)^{-1/2}$ Is this the correct answer?

Thanks
Jason

**Chris L T521** · June 21st 2009, 09:50 PM

Originally Posted by Darkhrse99

I knew to use the power rule, but I was coming up with
$1/2(x-1)^-1/2$ Is this the correct answer?

The derivative (by power rule & chain rule) is $1-\left(x-1\right)^{-1/2}=1-\frac{1}{\sqrt{x-1}}$ , where the derivative of $\sqrt{x-1}$ is $\frac{1}{2}\left(x-1\right)^{-1/2}=\frac{1}{2\sqrt{x-1}}$

**Darkhrse99** · June 22nd 2009, 09:57 PM

Originally Posted by Chris L T521

The derivative (by power rule & chain rule) is $1-\left(x-1\right)^{-1/2}=1-\frac{1}{\sqrt{x-1}}$ , where the derivative of $\sqrt{x-1}$ is $\frac{1}{2}\left(x-1\right)^{-1/2}=\frac{1}{2\sqrt{x-1}}$

How did you know to use the chain rule?

**Krizalid** · June 23rd 2009, 01:15 PM

chain rule is present at any moment.

for example if you want to differentiate a simple function as $x^2,$ then that is $2x\cdot(x)'=2x,$ but that it's quite simple and then irrelevant to write, but having $\sqrt{x-1}$ we can think this as the composition by letting $f(x)=\sqrt x$ and $g(x)=x-1,$ so by differentiating $f\big(g(x)\big)$ we get $f'\big(g(x)\big)\cdot g'(x).$ According to this we have that $\sqrt{x-1}$ can be expressed as $h(x)=f\big(g(x)\big)=\sqrt{x-1},$ and then by differentiating we get (following up on what I said before) $h'(x)=\frac1{2\sqrt{x-1}}\cdot(x-1)'=\frac1{2\sqrt{x-1}}.$

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