calc: Differentiation
11.5.1 Differentiation
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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’; 
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)’. 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).
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