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Please, how to evaluate dh(g(x))/dg(f(x)) by the following definitions:

f[x_] := x^4 - 1
g[f[x] _] := Log[f[x] + Sqrt[f[x]^2 - 1]]/Sqrt[f[x]^2 - 1]
h[g[x] _] := 1 + g[x]
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  • $\begingroup$ Define g as g[x_] := ... and h as h[x_] := .... Then D[h[x], x] seems to work. $\endgroup$
    – Chris K
    Commented May 21, 2019 at 19:51
  • $\begingroup$ but differentiation wrt g(f(x)) is required @ChrisK $\endgroup$
    – Emad Raslan
    Commented May 21, 2019 at 20:45
  • $\begingroup$ Sorry I overlooked that. I'm not sure what dh(g(x))/dg(f(x)) means, so hopefully someone else answers! $\endgroup$
    – Chris K
    Commented May 21, 2019 at 20:50
  • $\begingroup$ many thanks @Chris K $\endgroup$
    – Emad Raslan
    Commented May 21, 2019 at 20:51
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    $\begingroup$ With @ChrisK instructions on g and h then does not FullSimplify@D[g[f[x]], x] provide the answer you are seeking? $\endgroup$
    – Edmund
    Commented May 21, 2019 at 23:53

1 Answer 1

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Try the following code:

f[x_] := x^4 - 1;
g[x_] := Log[x + Sqrt[x^2 - 1]]/Sqrt[x^2 - 1];
h[x_] := 1 + x;
D[h@g@x, x] / D[g@f@x, x] // FullSimplify
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  • $\begingroup$ Thank you @Somos . It works $\endgroup$
    – Emad Raslan
    Commented May 22, 2019 at 22:04

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