The TangentLine(f, var) command computes the line tangent to the curve f at the point specified in var. The curve can be entered as a free or position Vector, a Vector-valued procedure, a scalar expression, or a scalar-valued procedure. If f is entered as a procedure, then it must have precisely one parameter.

If f is a scalar expression, then it must be given in terms of a variable from the current coordinate system, and the current coordinate system must be two-dimensional (see VectorCalculus[SetCoordinates]). If Maple cannot infer the parameter of the curve, an error will be raised, requesting a parameter variable name to be provided in the second argument. The expression is put into the Vector form using a new line parameter, usually t. For example, if the coordinate system is cartesian[x,y], and f is a function of y, then f is converted to a Vector of the form <f, t>.

If var is an equation, the left-hand side is the parameter of the curve and the right-hand side is the point in question. If var is an expression, the parameter of the curve will be inferred from f.

The following circumstances require the second argument to be of the form name=algebraic, giving the parameter name of the curve:

f is a Vector and more than one indeterminate is found in the components of f;

f is a scalar expression and is given in terms of more than one coordinate variable name from the current coordinate system;

f is a scalar expression and does not depend on any coordinate variable name from the current coordinate system; or

f is a scalar-valued procedure and its parameter name is not a coordinate variable name from the current coordinate system.

If the input curve is a Vector or a scalar, the output is a position Vector. If the input curve is a procedure, the output is a procedure.

If the current coordinate system is not indexed by its coordinate variable names, then the default names will be used, should they be requested during computation. For example, if the current coordinate system is cartesian (in two dimensions), then if required, the default names x and y will be used as the coordinate variable names.