Translate this page into:


Welcome to Nolver's Science Corner! If you would like to read my blog, visit my website on Nolver's Cozy Corner. This page is dedicated to people who are into scientific subjects. Don't hesitate to be a member of this page!

For tags, visit my website on Nolver's Pretty Tags, for PSP tubes, visit my page on Nolver's Cool PSP Tubes. For my personal blog, visit my page on Nolver's Room.

**Always scroll down to read the actual post**

©2013-2017 Nolver's Science Corner. Images with my watermark are copyrighted by me, so I reserve all rights. It is not allowed to edit or to modify the pictures in any way. It is not allowed to use the images/backgrounds from the layout and from the welcome box. Please respect my work.

Saturday, September 14, 2013


In calculus, a branch of mathematics, the derivative is a measure 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 a moving object with respect to time is the object's instantaneous velocity.

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. Informally, the derivative is the ratio of the infinitesimal change of the output over the infinitesimal change of the input producing that change of output. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function.

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

Differentiation and the derivative

Differentiation is a method to compute the rate at which a dependent output 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 that y is a function of x. This functional relationship is often denoted y = f(x), where f denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures 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 divided by x is a straight line. In this case, y = f(x) = m x + b, for real numbers m and b, and the slope m is given by

m=\frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x},

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in." This formula is true because

y + Δy = f(x + Δx) = m (x + Δx) + b = m x + m Δx + b = y + m Δx.

It follows that Δy = m Δx.

This gives an exact value for the slope of a straight line. If the function f is not linear (i.e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.
The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.

In Leibniz's notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written

 \frac{dy}{dx} \,\!

suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of y with respect to x", "d y by d x", or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)

The most common approach to turn this intuitive idea into a precise definition uses limits, but there are other methods, such as non-standard analysis.