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Thread: derivative of inverse function

  1. #1
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    derivative of inverse function

    prove that d/dx arctan x = 1/(1+x^2)

    the solution given by the lecturer...

    (f-1)'(x) = 1/f'(f-1(x)) ....... by theorem 1
    = 1/cos(arcsin x)
    = 1/sqrt(1-sin^2(arcsin x)
    = 1/sqrt(1-x^2)



    just wandering how theorem 1 is derived.... =p anyone can explain to me... need to understand why rather than memorizing the formula...
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by bobey View Post
    prove that d/dx arctan x = 1/(1+x^2)

    the solution given by the lecturer...

    (f-1)'(x) = 1/f'(f-1(x)) ....... by theorem 1
    = 1/cos(arcsin x)
    = 1/sqrt(1-sin^2(arcsin x)
    = 1/sqrt(1-x^2)



    just wandering how theorem 1 is derived.... =p anyone can explain to me... need to understand why rather than memorizing the formula...
    Assuming that $\displaystyle f(x)$ has an inverse $\displaystyle f^{-1}(x)$, we have that $\displaystyle f(f^{-1}(x))=x$, for all $\displaystyle x$.
    By the chain rule it follows from this that $\displaystyle f'(f^{-1}(x))\cdot (f^{-1})'(x)=1$, now divide both sides by $\displaystyle f'(f^{-1}(x))$.
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  3. #3
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    Hello, bobey!

    Prove that: .$\displaystyle \frac{d}{dx}(\arctan x) \:=\: \frac{1}{1+x^2}$
    The usual method goes something like this . . .


    Let: .$\displaystyle y \;=\;\arctan x$

    Then: .$\displaystyle \tan y \:=\:x$ .[1]


    Differentiate implicitly:

    . . $\displaystyle \sec^2y\,\frac{dy}{dx} \:=\: 1 \quad\Rightarrow\quad \frac{dy}{dx} \;=\;\cos^2\!y$ .[2]


    From [1], we have: .$\displaystyle \tan y \:=\:\frac{x}{1} \:=\:\frac{opp}{adj}$

    Then $\displaystyle y$ is an angle in a right triangle with: $\displaystyle opp = x,\;adj = 1$
    Pythagorus says: .$\displaystyle hyp \:=\:\sqrt{1+x^2}$
    . . Hence: .$\displaystyle \cos y \;=\;\frac{adj}{hyp} \;=\;\frac{1}{\sqrt{1+x^2}}$


    Therefore, [2] becomes: .$\displaystyle \frac{dy}{dx} \;=\;\left(\frac{1}{\sqrt{1+x^2}}\right)^2 \;=\;\frac{1}{1+x^2} $


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