Vectors and functions is *here* in pdf form.

The quotient of an *n*-dimensional vector variable or vector-valued function **x** with a scalar variable *t* equals

The quotient of a scalar variable *x* with an *n*-dimensional vector variable or vector-valued function **t** equals

Note: transpose these vectors for *numerator notation*.

The quotient of an *n*-dimensional vector variable or vector-valued function **x** with an independent *n*-dimensional vector variable or vector-valued function **t** equals

The derivative of an *n*-dimensional vector function **x** with respect to an independent scalar variable *t* equals

The directional derivative of a scalar function *x* with respect to an independent *n*-dimensional vector variable **t** equals

Note: transpose these vectors for *denominator notation*.

The directional derivative of an *n*-dimensional vector-valued function **x** with respect to an *n*-dimensional vector variable **t** equals

The matrix is a square Jacobian matrix, whose (*i, j*)^{th} entry is

The inverse function theorem states that the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the inverse function.