Determine the second derivative of each of the follwoing:

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Determine the second derivative of each of the follwoing:

Mathematics
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f(x)=sqrt x
f1(x)=x^1/2
\[ \sqrt{x}=x^{1/2} \]

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i know
but know what about the second derivative
what is your starting equation?
f(x)= sqrt x
wait but how are we goona get the second derivative
we are going to apply the power rule again.
We apply the power rule twice, that is all. (Want an example?)
yeah
Ok, just tell me what do you get for the derivative of \(x^{1/2}\), when you apply the power rule?
i goted it
Yeah, IrishBoy, lol. I indeed made the biggest mistake in the world. 9I guess the integration power rule got me mixed up just a bit)
The first derivative of \(\large\color{black}{ \displaystyle x^n }\) is given by the power rule: \(\large\color{black}{ \displaystyle \frac{d}{dx}\left[x^n\right]=nx^{n-1} }\) and then the second derivative of that would be: \(\large\color{black}{ \displaystyle \frac{d^2}{dx^2}\left[x^n\right]=\frac{d}{dx}\left[nx^{n-1}\right]=n(n-1)x^{n-2} }\) perhaps there are a few exceptions to this rule: \(n\ne 1\) and \(n\ne 0\) *++++++++++++++++++++++++++++++* \(\Large\color{black}{ \displaystyle f(x)=x^{\frac{1}{5}} }\) the first derivative, using the power rule is as follows: \(\Large\color{black}{ \displaystyle f'(x)=\left(\frac{1}{5}\right)x^{\frac{1}{5}-1} =\frac{1}{5}x^{-\frac{4}{5}}}\) Then, the second derivative you would find by differentiating f'(x) again, using the power rule. \(\Large\color{black}{ \displaystyle f''(x)=\left(\frac{1}{5}\right)\left(-\frac{4}{5}\right)x^{-\frac{4}{5}-1}}\) and this simplifies to: \(\Large\color{black}{ \displaystyle f''(x)=-\frac{4}{5}x^{-\frac{9}{5}}}\)

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