Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value)
Derivative of a function has a very natural geometric and physical interpretation: it corresponds to slope of the tangent line and to instantaneous velocity. In applications, it usually describes the rate of change of a physical variable.
Basic techniques used for computing the derivative of a given function are
- the definition of the derivative,
- the product rule and quotient rule and
- the chain rule.
It is useful to know the derivatives of elementary functions. This tag is intended for questions on the evaluation of derivatives.
Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
