In this investigation we will explain the term derivative.
Derivative is in calculus (a branch of mathematics) the derivative is ameasure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example,the derivative of the position of an object with respect to time is the object's instantaneous velocity.
Differentiation is a method to compute the rate at which a dependentoutput y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means thaty is a function of x. This functional relationship is often denoted y = ƒ(x), where ƒ denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivativemeasures the slope of this graph at each point.
The simplest case is when y is a linear function of x, meaning that the graph of y against x is a straight line. In this case, y = ƒ(x) = m x + b, for realnumbers m and b, and the slope m is given by
The derivative as a function
Let ƒ be a function that has a derivative at every point a in the domain of ƒ. Because every point a has a derivative,there is a function that sends the point a to the derivative of ƒ at a. This function is written f′(x) and is called the derivative function or the derivative of ƒ. The derivative of ƒ collects all thederivatives of ƒ at all the points in the domain of ƒ.
Sometimes ƒ has a derivative at most, but not all, points of its domain. The function whose value at a equals f′(a) whenever f′(a) is defined...