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MTALHAHASSAN2

  • one year ago

Determine the second derivative of each of the follwoing:

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  1. MTALHAHASSAN2
    • one year ago
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    f(x)=sqrt x

  2. MTALHAHASSAN2
    • one year ago
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    f1(x)=x^1/2

  3. thomas5267
    • one year ago
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    \[ \sqrt{x}=x^{1/2} \]

  4. MTALHAHASSAN2
    • one year ago
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    i know

  5. MTALHAHASSAN2
    • one year ago
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    but know what about the second derivative

  6. MTALHAHASSAN2
    • one year ago
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    @IrishBoy123

  7. IrishBoy123
    • one year ago
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    what is your starting equation?

  8. MTALHAHASSAN2
    • one year ago
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    f(x)= sqrt x

  9. MTALHAHASSAN2
    • one year ago
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    @SolomonZelman

  10. MTALHAHASSAN2
    • one year ago
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    wait but how are we goona get the second derivative

  11. SolomonZelman
    • one year ago
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    we are going to apply the power rule again.

  12. SolomonZelman
    • one year ago
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    We apply the power rule twice, that is all. (Want an example?)

  13. MTALHAHASSAN2
    • one year ago
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    yeah

  14. SolomonZelman
    • one year ago
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    Ok, just tell me what do you get for the derivative of \(x^{1/2}\), when you apply the power rule?

  15. MTALHAHASSAN2
    • one year ago
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    i goted it

  16. SolomonZelman
    • one year ago
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    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)

  17. SolomonZelman
    • one year ago
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    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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