What exactly is a derivative in calculus?

Question

I know how to find the derivative and its application (i.e. finding the slope of tangent line) but what exactly is it?

Answer 1

THE DERIVATIVE
The rate of change of a function
at a specific point
THE SLOPE of a straight line indicates the rate of change of y with respect to x, as we move from a point A on the line to a point B; so many units of y for each unit of x. (Topic 8 of Precalculus.) For, as we move from A to B the coordinates change.The slope is the number

Δy
Δx = = Change in y-coordinate
Change in x-coordinate .
The rate of change -- the slope -- of a straight line is constant. A straight line has one and only one slope.
If x represents time, for example, and y represents distance, then a straight line graph that relates them indicates constant speed. 45 miles per hour, say -- at every moment of time.

The slope of a tangent line to a curve

Calculus, however, is concerned with rates of change that are not constant.
If this curve represents distance versus time, then the rate of change — the speed — at each moment of time is not constant. Calculus asks this question: "What is the rate of change at exactly the point P ?" The answer will be the slope of a line tangent to the curve at precisely that point. And the method for finding that slope -- that number -- was the remarkable discovery by both Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716). That method is the fundamental procedure of differential calculus.A secant is a straight line that cuts a curve. (A tangent is a straight line that just touches a curve.) Hence, draw a secant line cutting the curve at points P and Q. The slope of the secant is the average rate of change between those two points.

Answer 2

You know how you can't know the speed something is going and it's exact location at the same time. The derivative is a way of finding like an aproximate speed at a set time, that is why when you choose a slope you try to find two points that are very close together. There are some very useful aplications for the derivitave but I can't think of them right now.

Answer 3

possibly a greater perfect question is, what good does looking a spinoff of a function do for me? nicely, derivatives are clever for looking out how something is changing. despite in case you look at a graph, we could say some graph showing a vehicle's place over the years, it particularly is clever to comprehend how briskly the vehicle grew to become into going at diverse situations, enable's say in case you had a rushing cost ticket gadget. So, in case you have some function f(t) ,*t for time*, that shows the situation of a vehicle, in case you're taking the spinoff, you have some function f ' (t) that shows the speed at each and every element in time. Now in case you needed to comprehend if the vehicle grew to become into rushing , you at the instant have a thank you to confirm.

Answer 4

the first derivative is the slope of a curve at a given point on the curve.

Answer 5

In general, "a" calculus is an abstract theory developed in a purely formal way.

"The" calculus, more properly called analysis (or real analysis or, in older literature, infinitesimal analysis) is the branch of mathematics studying the rate of change of quantities (which can be interpreted as slopes of curves) and the length, area, and volume of objects. The calculus is sometimes divided into differential and integral calculus, concerned with derivatives
d/(dx)f(x)

and integrals
intf(x)dx,

respectively.

Answer 6

Go search it in google.com. People (including me) are too lazy to explain it to you here.

Answer 7

what ever you want to make it "wink wink"

Answer 8

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