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Math Help - Find The Derivative of the Function x^2√(1-x^2)

  1. #1
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    Find The Derivative of the Function x^2√(1-x^2)

    f(x)=(x^2)(√(1-x^2))

    f'(x)=(2x)(√(1-x^2))+(x^2)(1/2)(√(1-x^2))^-1)(-2x)

    f'(x)=(2x)((1-x^2)^1/2)+(x^2)(1/2)(√(1-x^2)^-1)(-2x)

    f'(x)=√(4x^2-4x^4)+(1/2)(-2x^3)(1/(√(1-x^2))

    umm i'm lost, i probably didn't even get the above right (sorry the format sucks pretty bad)



    never mind i found a picture
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  2. #2
    Super Member Quacky's Avatar
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    Re: Find The Derivative of the Function x^2√(1-x^2)

    I find your notation very difficult to read.

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

    I'd use the product rule.

    Let u=x^2

    then \frac{du}{dx}=?

    Let v=(1-x^2)^{\frac{1}{2}}

    then \frac{dv}{dx}=?

    You should be able to proceed from here.
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  3. #3
    Member sbhatnagar's Avatar
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    Lightbulb Re: Find The Derivative of the Function x^2√(1-x^2)

    x^2\sqrt{1-x^2}=\sqrt{x^4-x^6}

    and

    \frac{d}{dx}(\sqrt{f(x)})=\frac{f'(x)}{2\sqrt{f(x)  }}
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  4. #4
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    Re: Find The Derivative of the Function x^2√(1-x^2)

    Quote Originally Posted by Deo3560 View Post
    f(x)=(x^2)(√(1-x^2))

    f'(x)=(2x)(√(1-x^2))+(x^2)(1/2)(√(1-x^2))^-1)(-2x)

    f'(x)=(2x)((1-x^2)^1/2)+(x^2)(1/2)(√(1-x^2)^-1)(-2x)
    f'(x)=(2x)(1-x^2)^{1/2}-(x^2)(1/2)(1-x^2)^{-1/2}(-2x)

    It is right up to here, and may be simplified:

    f'(x)=(2x)(1-x^2)^{1/2}+\frac{x^3}{(1-x^2)^{1/2}}

    It is easier to proceed by taking out a factor of x \sqrt{1-x^2)

    f'(x) =x(1-x^2)^{1/2}\left[2 -\frac{x^2}{1-x^2}  \right]

    That square bracket may be simplified further if you wish.

    CB
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