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Whilst creating a Java program I began to question the inbuilt functions that it provides.  One however wasn't present.  This was differentiation.  I began to question how a computer would go about computing differentiation.. any answers?

computing the derivative of a function, whereby the answer is a function, would require some complex inbuilt table for the program to follow and would be very difficult. Getting the computer to approximate the slope of a tangent line at a specific point, however, could be made arbitrarily close approximation by letting the computer take arbitrarily small values of h for [f(x+h)-f(x)]/h. You would get strange results, however, when 'differentiating' at points where the function has no defined derivative. this would require another table to correct, but a smaller one.

To do it numerically, a variety of schema can be amployed, though simple finite differences is most common.

Closed form differentiation is larely done by computers much like we to.  They parse the problem up, perform the tasks, and try to reduce it to a compact form.  I doubt anything in Java is very sophisticated, nothing like Mathematica.  If it was to be added, it would n't be for something potentially computation intensive like closed form differentiation or integration.

Symbolic differentiation is comparatively easy, and polynomial differentiation is often assigned as a programming project in a second-term programming class.  Full-blown differentiation is of course more challenging, but since the algorithm is clear cut (remember, every first semester calculus student learns to differentiate, in principle, every elementary function), it is not that hard to implement.  The hard part is actually "simplifying" answers that get spewed out.  By contrast, symbolic integration is much harder.