differentiate 2^(x^2)

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  1. anonstudent1's Avatar
    1 29-05-2012  23:48
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    differentiate 2^(x^2)
    I know if y=a^x dy/dx= (a^x)lna

    So for 2^(x^2) would the right answer to this be 2^(x^2)ln2 ?
  2. Rennit's Avatar
    2 29-05-2012  23:53
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    Re: differentiate 2^(x^2)
    The easiest way to think about this is to first rewrite x^(x^2) as e^(kx^2). Then simply it would be 2kxe^(kx^2)
  3. raheem94's Avatar
    3 29-05-2012  23:55
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    Re: differentiate 2^(x^2)
    (Original post by anonstudent1)
    I know if y=a^x dy/dx= (a^x)lna

    So for 2^(x^2) would the right answer to this be 2^(x^2)ln2 ?
    No.

    Example:

    y = 3^{x^2}

    Take the natural log of both sides

    \displaystyle lny = ln3^{x^2} \implies lny = x^2 ln3

    Now use implicit differentiation,

    \displaystyle \frac1{y} \frac{dy}{dx} = 2xln3 \implies \frac{dy}{dx} = 2xyln3 = 2x 3^{x^2} ln3
  4. latentcorpse's Avatar
    4 29-05-2012  23:57
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    Re: differentiate 2^(x^2)
    write it as e^{\ln{2^{x^2}}}

    then differentiate using chain and product rules
  5. Раскольников's Avatar
    5 30-05-2012  00:01
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    Re: differentiate 2^(x^2)
    http://www.wolframalpha.com/input/?i...%5E%28x%5E2%29

    In general a good site for checking your answers.
  6. f1mad's Avatar
    6 30-05-2012  00:10
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    Re: differentiate 2^(x^2)
    Let y= 2^(x^2)

    Take ln of both sides.
  7. anonstudent1's Avatar
    7 30-05-2012  00:33
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    Re: differentiate 2^(x^2)
    (Original post by Раскольников)
    http://www.wolframalpha.com/input/?i...%5E%28x%5E2%29

    In general a good site for checking your answers.
    Very helpful thanks!
  8. anonstudent1's Avatar
    8 30-05-2012  00:33
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    Re: differentiate 2^(x^2)
    (Original post by raheem94)
    No.

    Example:

    y = 3^{x^2}

    Take the natural log of both sides

    \displaystyle lny = ln3^{x^2} \implies lny = x^2 ln3

    Now use implicit differentiation,

    \displaystyle \frac1{y} \frac{dy}{dx} = 2xln3 \implies \frac{dy}{dx} = 2xyln3 = 2x 3^{x^2} ln3
    Thank you!
  9. anonstudent1's Avatar
    9 30-05-2012  00:37
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    Re: differentiate 2^(x^2)
    (Original post by raheem94)
    No.

    Example:

    y = 3^{x^2}

    Take the natural log of both sides

    \displaystyle lny = ln3^{x^2} \implies lny = x^2 ln3

    Now use implicit differentiation,

    \displaystyle \frac1{y} \frac{dy}{dx} = 2xln3 \implies \frac{dy}{dx} = 2xyln3 = 2x 3^{x^2} ln3
    Apparently the answer is 3^(x^2)xln9 according to this site http://www.wolframalpha.com/input/?i...%5E%28x%5E2%29
  10. raheem94's Avatar
    10 30-05-2012  00:47
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    Re: differentiate 2^(x^2)
    (Original post by anonstudent1)
    Apparently the answer is 3^(x^2)xln9 according to this site http://www.wolframalpha.com/input/?i...%5E%28x%5E2%29
    Both are correct.

    \displaystyle 2x 3^{x^2} ln3 = x 3^{x^2} \times 2ln3 = x3^{x^2} \times ln3^2 = x3^{x^2}ln9
  11. anonstudent1's Avatar
    11 30-05-2012  00:57
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    Re: differentiate 2^(x^2)
    y=2^(x^2)
    lny=x^2 ln2
    1/y dy/dx=2x ln 2
    dy/dx=2^(x^2) 2x ln2

    But the answer is 2^(x^(2+1)) x ln2

    Can anyone tell me how I get from one to the other, or if my answer is wrong?
  12. raheem94's Avatar
    12 30-05-2012  01:04
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    Re: differentiate 2^(x^2)
    (Original post by anonstudent1)
    y=2^(x^2)
    lny=x^2 ln2
    1/y dy/dx=2x ln 2
    dy/dx=2^(x^2) 2x ln2

    But the answer is 2^(x^(2+1)) x ln2

    Can anyone tell me how I get from one to the other, or if my answer is wrong?
    The answer should be 2^{x^2+1} x ln2 .

    You have got 2^{x^2} 2x ln2 = 2^1 \times 2^{x^2} \times xln2 = 2^{x^2+1}xln2
  13. anonstudent1's Avatar
    13 30-05-2012  01:09
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    Re: differentiate 2^(x^2)
    (Original post by raheem94)
    The answer should be 2^{x^2+1} x ln2 .

    You have got 2^{x^2} 2x ln2 = 2^1 \times 2^{x^2} \times xln2 = 2^{x^2+1}xln2
    Yh i see. Thank you, you have been very helpful!
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