∂/∂x

This operation determines the derivative of the expression against the provided symbol. Most operations in the calculator have known derivatives so expressions which are some combination of these built-in operations can be determined.

Derivatives can appear within an expression also but the syntax is a bit different. A derivative of SIN(X) against X looks like:

'∂X(SIN(X))'

So, the symbol to evaluate against appears after the derivative operation and the expression to evaluate appears inside brackets. When determining the derivative of a function off the stack by pressing the derivative button, the derivative is fully evaluated. That means that no "∂" operation will appear in the result. But, if you evaluate an expression with a "∂" operation in it, a single step in the derivative will be performed. So, in the above example, if you do an "EVAL" on that expression, the result will be:

'COS(X)*∂X(X)'

And then after another "EVAL":

'COS(X)*1'

For custom operations created by the user or for built-in operations with no known derivative, the calculator looks for a specific symbol which you can use to provide a derivative. For example, if you have created an operation called FOOBAR which takes two arguments, then you can try to evaluate an expression like this:

'∂X(FOOBAR(X,X))'

After evaluating this, you will get:

'derFOOBAR(X,X,∂X(X),∂X(X))'

The calculator prefixes the operation with "der". The "der" operation takes twice as many arguments as the non-"der" operation. In the case of an operation like FOOBAR which takes two arguments, derFOOBAR's first argument is the first argument to FOOBAR and the second is the second. The third argument to derFOOBAR is the derivative of the first argument and the fourth argument is the derivative of the second.

As a way of an example, let's imagine that SIN() did not have a known derivative on the calculator (it does, but if it didn't...). We can provide the derivative by setting 'derSIN' to the following program:

≪ → x dx 'COS(x)*dx' ≫

This program encapsulates the proper derivative for SIN() and an application of the chain rule which says that the derivative of SIN(f(x)) with respect to x is COS(x) * f'(x). Using similar patterns, you can add derivatives for your custom operations.

NOTE: Because Halcyon Calc Lite does not include support for programs, you cannot define derivatives for custom operations on the free version of Halcyon Calc.