Differentiation
---------------

The `a d' (`calc-derivative') [`deriv'] command computes the
derivative of the expression on the top of the stack with respect to
some variable, which it will prompt you to enter.  Normally, variables
in the formula other than the specified differentiation variable are
considered constant, i.e., `deriv(y,x)' is reduced to zero.  With the
Hyperbolic flag, the `tderiv' (total derivative) operation is used
instead, in which derivatives of variables are not reduced to zero
unless those variables are known to be "constant," i.e., independent
of any other variables.  (The built-in special variables like `pi' are
considered constant, as are variables that have been declared `const';
See Declarations.)

With a numeric prefix argument N, this command computes the Nth
derivative.

When working with trigonometric functions, it is best to switch to
radians mode first (with `m r').  The derivative of `sin(x)'
in degrees is `(pi/180) cos(x)', probably not the expected
answer!

If you use the `deriv' function directly in an algebraic formula, you
can write `deriv(f,x,x0)' which represents the derivative of `f' with
respect to `x', evaluated at the point `x=x0'.

If the formula being differentiated contains functions which Calc does
not know, the derivatives of those functions are produced by adding
primes (apostrophe characters).  For example, `deriv(f(2x), x)'
produces `2 f'(2 x)', where the function `f'' represents the
derivative of `f'.

For functions you have defined with the `Z F' command, Calc expands
the functions according to their defining formulas unless you have
also defined `f'' suitably.  For example, suppose we define `sinc(x) =
sin(x)/x' using `Z F'.  If we then differentiate the formula `sinc(2
x)', the formula will be expanded to `sin(2 x) / (2 x)' and
differentiated.  However, if we also define `sinc'(x) = dsinc(x)',
say, then Calc will write the result as `2 dsinc(2 x)'.  *Note
Algebraic Definitions::.

For multi-argument functions `f(x,y,z)', the derivative with respect
to the first argument is written `f'(x,y,z)'; derivatives with respect
to the other arguments are `f'2(x,y,z)' and `f'3(x,y,z)'.  Various
higher-order derivatives can be formed in the obvious way, e.g.,
`f''(x)' (the second derivative of `f') or `f''2'3(x,y,z)' (`f'
differentiated with respect to each argument once).