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Mathematics Calculus Wolfram Language

Finding the derivative of InverseFunction[ ] ?

袁旭龙

Posted 3 years ago

POSTED BY: 袁旭龙

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 3 years ago

You can start with the basic relation `f[(f^-1)[x]] == x` and take its derivative: f = -((Sqrt[#1] (-E^(2 C[1]) + #1) + E^(3 C[1]) ArcSin[E^-C[1] Sqrt[#1]] Sqrt[1 - E^(-2 C[1]) #1])/Sqrt[E^(2 C[1]) - #1]) &; basicRelation = (f[fInverse[t + C[2]]] == t + C[2]); Solve[D[basicRelation, t], fInverse'[t + C[2]]]

You can start with the basic relation f[(f^-1)[x]] == x and take its derivative:

f = -((Sqrt[#1] (-E^(2 C[1]) + #1) +
        E^(3 C[1]) ArcSin[E^-C[1] Sqrt[#1]]
         Sqrt[1 - E^(-2 C[1]) #1])/Sqrt[E^(2 C[1]) - #1]) &;
basicRelation = (f[fInverse[t + C[2]]] == t + C[2]);
Solve[D[basicRelation, t], fInverse'[t + C[2]]]

POSTED BY: Gianluca Gorni

袁旭龙

Posted 3 years ago

Thank you! By the way, I have another problem (not related to the above one) to ask you: I used the NDSolve to solve a set of ordinary differential equations in a very narrow interval of the independent variable(say t) and it failed. Actually I managed to compute the values of functions in the equations at each t with very high working precision, but the values of functions weren't calculated with such high working precision in the NDSolve solving. It seemed I was solving at a single point. When I made the interval large, it worked. How can I solve the problem? Need I write my own differential equation solving program instead of NDSolve?

POSTED BY: 袁旭龙

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 3 years ago

Perhaps you can rescale the parameter `t` in the equation so that it ranges over a larger interval.

POSTED BY: Gianluca Gorni

袁旭龙

Posted 3 years ago

I think I tried, the interval got larger though it was still tiny compared with the enormous length from the origin to the target interval, which also appeared in the equations. Moreover, it sounds difficult to deal it with very high working precision. What book or website do you recommend me to read for rescaling tricks?

POSTED BY: 袁旭龙

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 3 years ago

You can always rescale the independent variable, for example this way: independentVariableChange = (at + b == x) dependentVariableChange = (g[t] == f[x]) dependentVariableChange /. Solve[independentVariableChange, x][[1]] % /. Solve[independentVariableChange, t][[1]] changeOfVariables = f -> (Function[x, g[t]] /. Solve[independentVariableChange, t][[1]]) originalDiffEq = f''[x] - f'[x] == xf[x] originalDiffEq /. changeOfVariables % /. Solve[independentVariableChange, x][[1]]

You can always rescale the independent variable, for example this way:

independentVariableChange = (a*t + b == x)
dependentVariableChange = (g[t] == f[x])
dependentVariableChange /. Solve[independentVariableChange, x][[1]]
% /. Solve[independentVariableChange, t][[1]]
changeOfVariables = 
 f -> (Function[x, g[t]] /. Solve[independentVariableChange, t][[1]])
originalDiffEq = f''[x] - f'[x] == x*f[x]
originalDiffEq /. changeOfVariables
% /. Solve[independentVariableChange, x][[1]]

POSTED BY: Gianluca Gorni

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