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Math Help - Inverse Trig Derivative Problem about ArcSin

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    Inverse Trig Derivative Problem about ArcSin

    Find the Derivative

    y = 2 \arcsin \sqrt{1 - 2x}

    using the formula for derivative of arcsin:

     \frac{d}{dx}\sin^{-1}

    and

    u = (\frac{1}{\sqrt{1 - u^{2}}})(\frac{du}{dx})

    Step 1.

    y = 2[\frac{1}{\sqrt{1-(\sqrt{1-2x})^{2}}}] [\frac{1}{2}(1-2x)^{-1/2}](-2)

    Step 2:

    Can't understand how they get from here (below this sentence) to the answer.

    y = (\frac{1}{\sqrt{1-1+2x}})(\frac{1}{\sqrt{1-2x}})(-2)

    Answer:

    y = \frac{-2}{\sqrt{2x - 4x^{2}}}
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    fkf
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    Re: Inverse Trig Derivative Problem about ArcSin

    Multiplying sqrt(a)*sqrt(b) = sqrt(a*b)

    In this case we have the denominator
    sqrt(1-1+2x)*sqrt(1-2x) = sqrt(2x)*sqrt(1-2x) = sqrt((2x)*(1-2x)) = sqrt(2x-4x^2)
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    Re: Inverse Trig Derivative Problem about ArcSin

    Quote Originally Posted by Jason76 View Post
    Can't understand how they get from here (below this sentence) to the answer.

    y = (\frac{1}{\sqrt{1-1+2x}})(\frac{1}{\sqrt{1-2x}})(-2)

    Answer:

    y = \frac{-2}{\sqrt{2x - 4x^{2}}}
    The notation in equation 3 was a little confusing, but no matter. It's clear what you meant.

    Note:
    \sqrt{a} \cdot \sqrt{b} = \sqrt{a}{b}

    Here we have a = 2x and b = 1 - 2x. Can you finish now?

    -Dan

    Ya beat me fkf!
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    Re: Inverse Trig Derivative Problem about ArcSin

    Quote Originally Posted by topsquark View Post
    The notation in equation 3 was a little confusing, but no matter. It's clear what you meant.

    Note:
    \sqrt{a} \cdot \sqrt{b} = \sqrt{a}{b}

    Here we have a = 2x and b = 1 - 2x. Can you finish now?

    -Dan

    Ya beat me fkf!
    Yes, thanks a lot. But I'm wondering, this was an example of one number inside of a radical being multiplied by two numbers in another radical. However, what if there were two numbers in one radical being multiplied by two in another (and similar patterns)?
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    Re: Inverse Trig Derivative Problem about ArcSin

    Quote Originally Posted by Jason76 View Post
    Yes, thanks a lot. But I'm wondering, this was an example of one number inside of a radical being multiplied by two numbers in another radical. However, what if there were two numbers in one radical being multiplied by two in another (and similar patterns)?
    If I am reading this correctly you are looking for \sqrt{cd} \cdot \sqrt{ef} = \sqrt{cdef}.

    It's similar to the equation I posted earlier. Just let a = cd and b = ed.

    If I am wrong about what you are asking, please let us know.

    -Dan
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