Derivatives

About Derivatives

Calculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science, and engineering, and it is critical to analysis in business and health as well. In this App Derivatives , we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions in this chapter so that we can then explore applications of these techniques .

In mathematics, 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). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Derivatives can 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.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus



Defining the Derivative :
The Derivative as a Function :
Differentiation Rules :
The derivative of a constant function is zero. The derivative of a power function is a function in which the power on x becomes the coefficient of the term and the power on x in the derivative decreases by 1. The derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative. The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.

Derivatives as Rates of Change :
In this App we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of change of a function. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics.

The Chain Rule :
Derivatives of Inverse Functions :
The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative. We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.

Implicit Differentiation :
We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations). By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.

Derivatives of Exponential and Logarithmic Functions :