Evaluate the difference quotient

Evaluate the difference quotient DEFAULT

Algebra Examples


Consider the differencequotientformula.

Find the components of the definition.

Evaluate the function at .

Replace the variable with in the expression.

Simplify the result.

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combinelike terms.

Simplify each term.

Add and .


Find the components of the definition.


Simplify the numerator.

Factor out of .

Reduce the expression by cancelling the common factors.

Cancel the common factor of .

Cancel the common factor.

Replace the variable with in the expression.

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Sours: https://www.mathway.com/examples/algebra/algebra-concepts-and-expressions/evaluating-the-difference-quotient

Difference Quotient is used to calculate the slope of the secant line between two points on the graph of a function, f. Just to review, a function is a line or curve that has only one y value for every x value. The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.

How to Use This Difference Quotient Calculator?

The procedure to use the difference quotient calculator is as follows:
Step 1: Enter two functions in the respective input field
Step 2: Now click the button ”Calculate Quotient” to get the result
Step 3: Finally, the difference quotient will be displayed in the new window

In calculus, the difference quotient is the formula used for finding the derivative which is the difference quotient between two points that are as close as possible which gives the rate of change of a function at a single point. The difference quotient was formulated by Isaac Newton.

Difference Quotient Calculator

This formula computes the slope of the secant line through two points on the graph of f. These are the points with x-coordinates x and x + h. The difference quotient is used in the definition of the derivative. First, plug (x + h) into your function wherever you see an x. Once you find f (x + h), you can plug your values into the difference quotient formula and simplify from there. In the third step, you use the subtraction sign to eliminate the parentheses and simplify the difference quotient.

Difference Quotient

In the formal definition of the difference quotient, you’ll note that the slope we are calculating is for the secant line. A secant line is just any line that passes between two points on a curve. We label these two points as x and (x +h) on our x-axis. Because we are working with a function, these points are labeled as f (x) and f (x + h) on our y-axis, respectively.

Difference Quotient: Definition, Formula

In simple terms, the difference quotient helps us find the slope when we are working with a curve. In the case of a curve, we cannot use the traditional formula of:

which is why we must use the difference quotient formula.


Difference Quotient Formula

The steps we take to find the difference quot are as follows:

Difference Quotient Formula
  • Plug x + h into the function f and simplify to find f(x + h).
  • Now that you have f(x + h), find f(x + h) – f(x) by plugging in f(x + h) and f(x) and simplifying.
  • Plug your result from step 2 in for the numerator in the difference quotient and simplify it.

which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (x+h)-x=h in this case).

Symmetric Difference Quotients

In mathematics, the difference quot is formulas that give approximations of the derivative of a function. There are a few different difference quots, and those are the one-sided difference quotients and the symmetric difference quot. They are all related, and one gives a better approximation than the others due to this relationship.

In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as:

symmetric difference quotient

The expression under the limit is sometimes called the symmetric difference quotient. A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point.

If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient.

The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point if the latter two both exist.

Neither Rolle’s theorem nor the mean value theorem hold for the symmetric derivative; some similar but weaker statements has been proved.

The modulus function

Graph of the modulus function. Note the sharp turn at x=0, leading to non-differentiability of the curve at x=0. The function hence possesses no ordinary derivative at x=0. The symmetric derivative, however, exists for the function at x=0.

For the modulus function, f(x)=|x|, we have, at x=0,

modulus function

where since h>0 we have  |h|=-(-h) . So, we observe that the symmetric derivative of the modulus function exists at x=0, and is equal to zero, even though its ordinary derivative does not exist at that point (due to a “sharp” turn in the curve at x=0).

Note in this example both the left and right derivatives at 0 exist, but they are unequal (one is -1 and the other is 1); their average is 0, as expected.

Difference Quotient Example

  • The difference quot for the function f(x)=3-x^2-x is:
Difference Quotient Example
  • The difference quot for the function  is:
Difference Quotient Example

The difference quot for the function  is:

Evaluate The Difference Quotient

Some practice problems for you; find the difference quot for each function showing all relevant steps in an organized manner (see examples).

  1. f(x)=3-7x
  2. k(t)=7x²+2
  3. z(x)=π
  4. s(t)=t³-t-9

How do you find the quotient?

Divide the dividend by the whole-number divisor to find the quot.

Multiply the divisor by a power of 10 to make it a whole number.

Multiply the dividend by the same power of 10. Place the decimal point in the quot.

Divide the dividend by the whole-number divisor to find the quot.

What is the derivative of SA?

A derivative is a contract between two or more parties whose value is based on an agreed-upon underlying financial asset (like a security) or set of assets (like an index). Common underlying instruments include bonds, commodities, currencies, interest rates, market indexes, and stocks.

What is a quotient example?

The answer after we divide one number by another. dividend ÷ divisor = quot. Example: in 12 ÷ 3 = 4, 4 is the quot. Division.

What is the formula of Quotient?

The quot rule is a formula for taking the derivative of a quot of two functions. If you have function f(x) in the numerator and the function g(x) in the denominator, then the derivative is found using this formula: In this formula, the d denotes a derivative.

What is a quotient function?

The quot function returns the integer portion of a division. Simple as that. quot(numerator, denominator) There are two arguments, the numerator is the dividend and the denominator is the divisor. … The Quot function returns 4 because the integer part of 4.5 is 4.

Sours: https://studyqueries.com/difference-quotient/
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If we are offered a feature and need to locate the incline at a factor. We can make an approximation by using the distinction ratio. To approximate the slope, we choose our x limitations on either side of the point. Picture the factor is right in the middle of the interval. The closer the period x limits are to the end, the much more accurately the distinction quotient will approximate the incline then. In other words, a narrower period = an extra accurate approximation. Let’s take a look at difference quotient basics.

As revealed previously in the instance. We additionally make use of the difference quotient to locate the typical rate of modification over a series of x values for a feature. Watch out for concerns that provide a function. As well as ask to find the ordinary incline or ordinary rate of change over a period or variety of x values. The difference ratio is specifically valuable when there are multiple indicate perform. Since it conserves time contrasted to using the incline formula.

A difference quotient is an approach for locating the ordinary rate of change of a function over a period. It calculates an approximated kind of a by-product. The distinction quotient given as:

Difference quotient

Where f(x) is the feature as well as h is the step size. This calculates the regular price of modification of the function f(x) over the interval [x, x + h] We use the difference ratio to our function, which develops a new feature of the variables x and also h.

About Calculus Derivatives

Acquired is the central principle of Calculus as well as is known for its numerous applications. To more great Math—by-product of a function at a point defined in 2 different means: geometric and also physical. Geometrically, the by-product of a feature at a particular value of its input variable. It is the incline of the line tangent to its graph through the provided point. It found by utilizing the slope formula or if offered a chart by attracting horizontal lines towards the input value under questions. If the graph has no break or dives. Then it is merely the y value representing the given x-value.

Read Also:Understand more about Negative Exponents

In Physics, the derivative described as a physical adjustment. It refers to the instantaneous rate of change in the speed of things concerning the shortest possible time it takes to take a trip a certain distance. In connection thereof, the by-product of a function at a factor in a Mathematical view describes. The rate of modification of the value of outcome variables as the values of its corresponding input variables obtains near zero. Put, if two carefully picked values are very near to the given point under concern, after that the by-product of the function at the point of inquiry is the ratio of the difference between the output values as well as their equivalent input values, as the denominator gets near zero (0).

precisely, the derivative of a function is a measurement of how a feature transforms relative to an adjustment of values in its input (independent) variable.

To discover the by-product of a function at a specific factor, do the complying with actions:

  1. Pick two values, very near to the given point, one from its left and also the other from its right.
  2. Solve for the matching outcome values or y values.
  3. Compare both values.
  4. If both values are the same or will about equal to the same number. Then it is the by-product of the function at that absolute worth of x (input variable).
  5. Making use of a table of values. If the values of y for those points to the right of the x worth under question is about equal to the y value. Approached by the y worths corresponding to the picked input values to the left of x. The value comes close to is the by-product of the function at x.
  6. Algebraically we can look for the acquired feature first. By taking the limit of the distinction quotient formula as the approaches no. Make use of the acquired trait to search for the by-product by changing the input variable with the given worth of x.
Sours: https://educationisaround.com/difference-quotient/
Pre-Calculus - Evaluate the difference quotient

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Quotient difference evaluate the

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Determining the Difference Quotient

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