# How to obtain the derivative analytically of a complicated function and evaluate it at 0

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Hi all,

I am new to the symbolic math toolbox. I am trying to differentiate a function and evaluate it at 0 (stable equilibrium). I got some sort of derivative from diff function, but if I try to evaluate at 0,I get NaN. I wish to do stability analysis around the equilibrium at 0. It is a discrete time model of a population. What I think I should do is to take the derivative with respect to F (the state variable, population density) and evaluate it at F = 0. And see if this derivative evaluated at 0 is less than 1, which indicates that the equilibrium is stable. I wish to derive an expression for the threshold value of D beyond which the extinction equilibrium is stable.

F(t) = (1-s0)*a*(1-exp(-r*F(t-1)/N))*N*(1-D)*(1-D/(D+s0*(a*(1-exp(-r*F(t-1)/N))*N/F(t-1))))

a = 0.62, r = 26, s0 = 0.5, N = 1.

This is what I did in Matlab

syms s0 a r F N D b

fun = @(s0,D,a,r,F,N) (1-s0)*a*(1-exp(-r*F/N))*N*(1-D)*(1-D/(D+s0*(a*(1-exp(-r*F/N))*N/F)))

diff(fun,F)

ans = - a*r*exp(-(F*r)/N)*(D - 1)*(D/(D - (N*a*s0*(exp(-(F*r)/N) - 1))/F) - 1)*(s0 - 1) - (D*N*a*((a*r*s0*exp(-(F*r)/N))/F + (N*a*s0*(exp(-(F*r)/N) - 1))/F^2)*(exp(-(F*r)/N) - 1)*(D - 1)*(s0 - 1))/(D - (N*a*s0*(exp(-(F*r)/N) - 1))/F)^2

myfun =@(s0,D,a,r,F,N) - a*r*exp(-(F*r)/N)*(D - 1)*(D/(D - (N*a*s0*(exp(-(F*r)/N) - 1))/F) - 1)*(s0 - 1) - (D*N*a*((a*r*s0*exp(-(F*r)/N))/F + (N*a*s0*(exp(-(F*r)/N) - 1))/F^2)*(exp(-(F*r)/N) - 1)*(D - 1)*(s0 - 1))/(D - (N*a*s0*(exp(-(F*r)/N) - 1))/F)^2 - 1

F_ = 0; N_ = 1; s0_ = 0.5; a_ = 0.62; r_ = 26;

myfun2 = @(D) myfun(s0_,D,a_,r_,F_,N_)

D = fzero(myfun2,0.8)

Then I got

Error using fzero (line 307)

Function value at starting guess must be finite and real.

Thank you very much for your help in advance!!

Etsuko

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### Answers (1)

Mahesh Pai
on 29 Mar 2017

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