# NDSolve of pre-defined function

I have defined function below want to calculate its Numerical differential using NDSolve.

x=A*Exp[-Log[y]^2];

f[x_]:= x*(1-x);


If i want to look at the change of $f[x]$ with respect to $x'[y]$ using NDSolve. what i have to do??

• You can format inline code and code blocks by selecting the code and clicking the {} button above the edit window. The edit window help button ? is useful for learning how to format your questions and answers. You may also find this meta Q&A helpful – Michael E2 Dec 7 '16 at 15:48
• do not use capital D as a symbol, it is reserved. Additionally you can not use bars | for absolute value, use Abs and nabla has no special meaning. – george2079 Dec 7 '16 at 16:51
• on top of the syntax issues your equation is algebraic in k so why are you trying to use NDSolve ? – george2079 Dec 7 '16 at 16:59
• y cannot be both a real number (independent variable) and a function head (dependent variable), which is what is happening in your code. There's a formula in calculus for y'[x] in terms of x'[y] -- you should probably try that. – Michael E2 Dec 7 '16 at 19:15
• @george2079 first of all thank u so much for bothering, I edited my question. can u still guide. – Amanullah Malik Dec 7 '16 at 19:30

x[y_] = A*Exp[-Log[y]^2]
f[y_] = x[y]*(1 - x[y])


The derivative of f[y] with respect to some function g[y] is simply

 D[f[y],y]/D[g[y],y]


so the derivative with respect to the derivative of x[y] is :

 result[y_]=D[f[y],y]/D[x[y],{x,2}] //FullSimplify


-(((1 - 2 A E^-Log[y]^2) y Log[y])/(-1 + Log[y] + 2 Log[y]^2))

checking..

 A = 1
z[y_] = D[x[y], y]
ParametricPlot[ {z[y], f[y]}, {y, .01, 10}, AspectRatio -> 1,
PlotRange -> {{-3/4, 2}, {-1/4, 1/2}},
Epilog -> (Arrow[{{z[#],
f[#]}, {z[#], f[#]} + .1 Normalize[{1, result[#]}]} ] & /@
Range[.05, 10, .1])]


• thanku so much, I think it works for me, and if I had $x[y]$ is so complicated that it's always on running. so preferably do $NDSolve$ than what changes i required. – Amanullah Malik Dec 7 '16 at 20:01
• maybe you want ND reference.wolfram.com/language/NumericalCalculus/ref/ND.html . ( I still don't get at all why you want to use NDSolve ) – george2079 Dec 7 '16 at 20:08
• actually my calculation is lengthy in someway, Further I have to define a function such as $k[y]$, which is sum of different different derivatives as given above, and at the end i have to integrated. If I do this analytically my mathematica running and no answer, but If I do it numerically within a range, it will be conclusive. I hope. – Amanullah Malik Dec 7 '16 at 20:20