f(x)=∫0x dt1+t3 and g(x) be the inverse of f(x). Then the value of  4g′′(x)(g(x))2  is __.

f(x)=0xdt1+t3 and g(x) be the inverse of f(x). Then the value of  4g′′(x)(g(x))2  is __.

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    Solution:

    y=f(x)x=f1(y)x=g(y)

    Given y=f(x)=0xdt1+t3

    dydx=11+x3 or dxdy=1+x3

    g(y)=1+g3(y)g′′(y)=3g2(y)g(y)21+g3(y) 2g′′(y)=3g2(y)g(y)1+g3(y)=3g2(y)1+g3(y)1+g3(y)=3g2(y)

    or 2g′′(y)=3g2(y)

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