The function depends returns true if any of the expressions contained in f are mathematically dependent on any of the names contained in x.

The expression f is mathematically dependent on the name x if it contains at least one instance of x which is not simply a dummy variable.  Examples of dummy variables are index variables in sums, products, integrals, and limits.  Dummy variables also appear in integral transforms.

If f is a complicated expression, then depends may not be able to determine that it is independent of x.  In such cases, f should simplified before depends is called.

There is a facility for the user to add additional functions to depends. If the function depends/myfunc is defined, then depends will use this function to check for dependence whenever it encounters myfunc.  The function depends/myfunc should accept as its parameters the arguments that myfunc expects in the same order, followed by a final parameter which will be a set containing all the names that dependence is being checked against.  Note that myfunc may be either a user-defined function or a standard library function.

Unrelated to the above, there is also a procedure parameter modifier, depends, that can be used in declaration to indicate that a parameter's type depends on the value of another parameter.

$\mathrm{depends}\left(\sin \left(x\right)+\cos \left(z\right)\,\left\{x\,y\right\}\right)$

$\mathrm{true}$

$\mathrm{depends}\left(\sin \left(x\right)+\cos \left(z\right)\,\left[x\,z\right]\right)$

$\mathrm{true}$

$\mathrm{depends}\left(\sin \left(x\right)+\cos \left(z\right)\,\left[y\,z\right]\right)$

$\mathrm{true}$

$\mathrm{depends}\left(\mathrm{int}\left(f\left(x\right)\,x=a..b\right)\,x\right)$

$\mathrm{false}$

$\mathrm{depends}\left(\mathrm{int}\left(f\left(x\right)\,x=a..b\right)\,a\right)$

$\mathrm{true}$

$\mathrm{depends}\left(\mathrm{limit}\left(\frac{1}{x}\,x=0\right)\,x\right)$

$\mathrm{false}$

$\mathrm{depends}\left(\mathrm{sum}\left(\frac{1}{x}\,x=1..n\right)\,x\right)$

$\mathrm{false}$

$\mathrm{depends}\left(\mathrm{sum}\left(\frac{1}{x}\,x=1..n\right)\,n\right)$

$\mathrm{true}$

$\mathrm{depends}\left(\mathrm{laplace}\left(f\left(t\right)\,t\,s\right)\,t\right)$

$\mathrm{false}$

$\mathrm{depends}\left({\sin \left(x\right)}^2+{\cos \left(x\right)}^2\,x\right)$

$\mathrm{true}$

$\mathrm{depends}\left(\mathrm{simplify}\left({\sin \left(x\right)}^2+{\cos \left(x\right)}^2\,\mathrm{trig}\right)\,x\right)$

$\mathrm{false}$