The derivative of a function is one of the basic concepts of mathematics. Together with the integral, derivative occupies a central place in calculus. The process of finding the derivative is called differentiation.
The inverse operation for differentiation is called integration. The derivative of a function at some point characterizes the rate of change of the function at this point. In the examples below, we derive the derivatives of the basic elementary functions using the formal definition of derivative. These functions comprise the backbone in the sense that the derivatives of other functions can be derived from them using the basic differentiation rules.
Copied or adapted from others: a highly derivative prose style. Linguistics A word formed from another by derivation, such as electricity from electric. Mathematics a. The limiting value of the ratio of the change in a function to the corresponding change in its independent variable. The instantaneous rate of change of a function with respect to its variable.
The slope of the tangent line to the graph of a function at a given point. Also called differential coefficientfluxion. Chemistry A compound derived or obtained from another and containing essential elements of the parent substance.
A financial instrument that derives its value from another more fundamental asset, as a commitment to buy a bond for a certain sum on a certain date.
All rights reserved. Chemistry chem a compound that is formed from, or can be regarded as formed from, a structurally related compound: chloroform is a derivative of methane.
Mathematics maths a. Psychoanalysis psychoanal an activity that represents the expression of hidden impulses and desires by channelling them into socially acceptable forms. Copyright, by Random House, Inc. In calculus, the slope of the tangent line to a curve at a particular point on the curve. Since a curve represents a function, its derivative can also be thought of as the rate of change of the corresponding function at the given point.
Donate Login Sign up Search for courses, skills, and videos. Calculus 1. Skill Summary Legend Opens a modal. Average vs. Newton, Leibniz, and Usain Bolt Opens a modal. Derivative as a concept Opens a modal. Derivative notation review Opens a modal. Derivative as slope of curve Opens a modal. Derivative as slope of curve Get 3 of 4 questions to level up! Secant lines. Slope of a line secant to a curve Opens a modal.
Secant line with arbitrary difference Opens a modal. Secant line with arbitrary point Opens a modal.
Secant line with arbitrary difference with simplification Opens a modal. Secant line with arbitrary point with simplification Opens a modal. Secant lines: challenging problem 1 Opens a modal. Secant lines: challenging problem 2 Opens a modal. Derivative definition. Formal definition of the derivative as a limit Opens a modal. Formal and alternate form of the derivative Opens a modal. Worked example: Derivative as a limit Opens a modal. Worked example: Derivative from limit expression Opens a modal.
Finding tangent line equations using the formal definition of a limit Opens a modal. Limit expression for the derivative of function graphical Opens a modal. Derivative as a limit Get 3 of 4 questions to level up! Estimating derivatives. Estimating derivatives Opens a modal.
Estimate derivatives Get 3 of 4 questions to level up! Differentiability and continuity Opens a modal. Differentiability at a point: graphical Opens a modal. Differentiability at a point: algebraic function is differentiable Opens a modal.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 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. 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. Differentiation is the action of computing a derivative. It is called the derivative of f with respect to x. If x and y are real numbersand if the graph of f is plotted against xthe derivative is the slope of this graph at each point. The simplest case, apart from the trivial case of a constant functionis when y is a linear function of xmeaning that the graph of y is a line.
The above formula holds because. If the function f is not linear i. Two distinct notations are commonly used for the derivative, one deriving from Gottfried Wilhelm Leibniz and the other from Joseph Louis Lagrange.
A third notation, first used by Isaac Newtonis sometimes seen in physics. In Leibniz's notationan infinitesimal change in x is denoted by dxand the derivative of y with respect to x is written.
The above expression is read as "the derivative of y with respect to x ", " dy by dx ", or " dy over dx ". The oral form " dy dx " is often used conversationally, although it may lead to confusion.
Lagrange's notation is sometimes incorrectly attributed to Newton. Newton's notation for differentiation also called the dot notation for differentiation places a dot over the dependent variable. That is, if y is a function of tthen the derivative of y with respect to t is.The slope of a straight line.
The slope of a tangent line to a curve. A secant to a curve. The difference quotient. The definition of the derivative. Differentiable at x. Notations for the derivative.
A simple difference quotient. The equation of a tangent to a curve. In a straight line, the rate of change -- so many units of y for each unit of x -- is constant, and is called the slope of the line. If this curve represents distance Y versus time Xthen the rate of change -- the speed -- at each moment of time is not constant.
And the method for finding that slope -- that number -- was the remarkable discovery by both Isaac Newton and Gottfried Leibniz That is the method for finding what is called the derivative. A tangent is a straight line that just touches a curve. A secant is a straight line that cuts a curve. Hence, consider the secant line that cuts the curve at points P and Q.Calculus: Derivatives 1 - Taking derivatives - Differential Calculus - Khan Academy
Then the slope of that secant is. And we will define the tangent at P to be the limit of that sequence of slopes. Topic 4 of Precalculus. Herethen, is the definition of the slope of the tangent line at P :. The slope of the tangent line at P is the limit of the change in the function the numerator divided by the change in the independent variable as that change approaches 0.
Since it will be derived from f xwe call it the derived function or the derivative of f x. Calculating and simplifying it is a fundamental task in differential calculus.
The difference quotient then becomes:. We now express the definition of the derivative as follows. By the derivative of a function f xwe mean the following limit, if it exists:.
We call that limit the function f ' x -- " f -prime of x " -- and when that limit exists, we say that f itself is differentiable at xand that f has a derivative. And so we take the limit of the difference quotient as h approaches 0. When that limit exists, that means that the difference quotient can be made as close to that limit -- " f ' x " -- as we please.
As for xwe are to regard it as fixed. In practice, we have to simplify the difference quotient before letting h approach 0. We have to express the numerator To sum up: The derivative is a function -- a rule -- that assigns to each value of x the slope of the tangent line at the point xf x on the graph of f x. It is the rate of change of f x at that point.The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways.
One is geometrical as a slope of a curve and the other one is physical as a rate of change. Historically there was and maybe still is a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful.
We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by. This concept of velocity may be extended to find the rate of change of any variable with respect to any other variable. For example, the volume of a gas depends on the temperature of the gas.
So in this case, the variables are V for volume as a function of T the temperature. Now we get to the hardest part. Since we can not keep on writing "Instantaneous Velocity" while doing computations, we need to come up with a suitable notation for it.
If we write dx for small, then we can use the notation. Recall that the graph of a function is a set of points that is xf x for x 's from the domain of the function f.
We may draw the graph in a plane with a horizontal axis usually called the x-axis and a vertical axis usually called the y-axis. Fix a point on the graph, say x 0f x 0. If the graph as a geometric figure is "nice" i.
Such a straight line is called the tangent line at the point in question.
Unit: Derivatives: definition and basic rules
The concept of tangent may be viewed in a more general framework. Note that the tangent line may not exist. We will discuss this case later on. One way to find the tangent line is to consider points xf x on the graph, where x is very close to x 0.
Then draw the straight-line joining both points see the picture below : As you can see, when x get closer and closer to x 0the lines get closer and closer to the tangent line. Since all these lines pass through the point x 0f x 0their equations will be determined by finding their slope: The slope of the line passing through the points x 0f x 0 and xf x where is given by.
Writing "m" for the slope of the tangent line does not carry enough information; we want to keep track of the function f x and the point x 0 in our notation. The common notation used is.This is such an important limit and it arises in so many places that we give it a name. We call it a derivative.
Here is the official definition of the derivative. So, we are going to have to do some work. In this case that means multiplying everything out and distributing the minus sign through on the second term. Doing this gives. After that we can compute the limit. This one is going to be a little messier as far as the algebra goes. However, outside of that it will work in exactly the same manner as the previous examples. First, we plug the function into the definition of the derivative.
Note that we changed all the letters in the definition to match up with the given function.
Also note that we wrote the fraction a much more compact manner to help us with the work. So, we will need to simplify things a little. In this case we will need to combine the two terms in the numerator into a single rational expression as follows. You do remember rationalization from an Algebra class right?
In an Algebra class you probably only rationalized the denominator, but you can also rationalize numerators. Remember that in rationalizing the numerator in this case we multiply both the numerator and denominator by the numerator except we change the sign between the two terms.
Introduction to Derivatives
So, cancel the h and evaluate the limit. We saw a situation like this back when we were looking at limits at infinity. We will have to look at the two one sided limits and recall that.
Derivatives will not always exist. The next theorem shows us a very nice relationship between functions that are continuous and those that are differentiable.
Note that this theorem does not work in reverse. Next, we need to discuss some alternate notation for the derivative. Because we also need to evaluate derivatives on occasion we also need a notation for evaluating derivatives when using the fractional notation. In these cases the following are equivalent. It is an important definition that we should always know and keep in the back of our minds.
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