Im looking for a  way to calculate a derivative in c++. An example of
code would be usefull to.

Hi shaneb2000!!

Here is what I found:
"A Guide to the Complex-Step Derivative Approximation" at University
of Toronto Institute for Aerospace Studies - Multidisciplinary Design
Optimization Laboratory website:
'The complex-step derivative approximation is a very convenient way of
estimating derivatives numerically. It is a simple and accurate means
of finding derivatives of a quantity calculated by an existing
algorithm.'
http://mdolab.utias.utoronto.ca/complex-step.html

At this site see:
"Complex-Step Derivative Approximation Official Guide: C++ Implementation":
http://mdolab.utias.utoronto.ca/c++.html

"Complex-Step Derivative Approximation Official Guide: C++ Template Example":
http://mdolab.utias.utoronto.ca/c++template.html

"THE CONNECTION BETWEEN THE COMPLEX-STEP DERIVATIVE APPROXIMATION AND
ALGORITHMIC DIFFERENTIATION" by Joaquim R. R. A. Martins, Peter
Sturdza and Juan J. Alonso:
http://mdolab.utias.utoronto.ca/documents/Martins2001.pdf

"Algorithmic Differentiation":
http://mdolab.utias.utoronto.ca/aer1415/aer1415_ho8.pdf


See also:
"Differential Arithmetic Library":
http://www.schmitteckert.com/boost/diffari/

See the Documentation page:
http://www.schmitteckert.com/boost/diffari/diff_ari.html


More references:
"Differentiation":
http://www.shodor.org/refdesk/Resources/Applications/Differentiation/index.php

"Micah Yoder's SEUL Page":
http://cran.mit.edu/~yoderm/



Search strategy:
"calculate derivatives" algorithm c++
"calculate derivatives" c++


I hope this helps you. If you find something unclear or incomplete,
please let me know using the clarification feature before rate this
answer.

Regards.
livioflores-ga

Methods for computing derivatives commonly fall into the symbolic or
numeric category.  By symbolic is meant some means of tying an input
expression to a resulting differentiated expression, possibly with
respect to one of several independent variables.  By numeric methods
are commonly meant difference quotients (based on the numerical
evaluation of the input expression), for which the derivative would be
a limit as some parameter (say h) tends to zero.

The latter category is often used as a simple illustration of
numerically ill-conditioned computations, for as the difference
parameter h approaches zero, the resulting numerical evaluation of the
difference quotient is prone to numerical inaccuracies dominated by
round-off errors (from numerical evaluation of the input expression).

Concern with this numerical inaccuracy (if it cannot be avoided by
symbolic computation) leads to a rule of thumb for choosing parameter
h so that its precision relative to the function's argument (the one
with respect to which the differentiation is being done) is about half
(in terms of digits of precision) of that available to the floating
point arithmetic used.

So we have a Goldilocks and the three bears situation.  A value of h
which is too large leads to a crude estimate of the derivative.  A
value of h which is too small leads to an estimate vulnerable to
round-off errors.  One seeks an intermediate value of h that is "just
right" for the intended purpose.

When higher-order derivatives (mixed or pure) are required, the issues
of symbolic vs. numeric computation and numeric instability are
ramified.  But second-order derivatives are commonly required in
optimization algorithms, at least in some approximated form. 
Discusssion of various strategies specific to this context can be
found by searching on terms like "conjugate search", "quasi-Newton",
and "BFGS".  One such reference is:

[Quasi-Newton (secant) methods]
http://www-personal.engin.umich.edu/~mepelman/teaching/IOE511/section7.pdf

Under mild continuity conditions the Hessian (matrix of second-order
partial derivatives) is symmetric, and one aspect of this subject is
the selection of approximation methods that preserve this symmetry.

regards, mathtalk-ga

Have you tried looking in the GSL for some optimized functions that do this?