Tuesday, 17 May 2022
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Difference quotient — all you need to know

We use the Difference Quotient to compute the slope of the secant line connecting two points on a function’s graph, f.

So, to recap, a function is a line or curve with only one y value for each x value. The difference quotient is a measure of the function’s average rate of change over an interval (in this case, an interval of length h).

Thus, the instantaneous rate of change is the limit of the difference quotient (i.e., the derivative).

The difference quotient is a formula used in calculus to find the derivative, which is the difference quotient between two points as close as possible that gives the rate of change of a function at a single point. Isaac Newton developed the difference quotient.

The difference quotient is sometimes regarded as the Newton quotient

Difference quotient

Difference quotient of a function

A function is an expression, rule, or law in mathematics that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

So, the difference quotient is a measure of the function’s average rate of change over an interval. The instantaneous rate of change is thus the limit of the difference quotient (i.e., the derivative).

Difference quotient formula derivation

Consider the function y = f(x) and a secant line that passes through two points on the curve (x, f(x)) and (x + h, f(x + h)). Using the slope formula, the slope of the secant line is as follows:

f(x + h) – f(x) [(x + h) – x] / f(x + h) – f(x) = [ h ]

(Because the slope of any straight line is equal to the change in y divided by the change in x.)

This is the difference quotient formula in action.

Hence, as h approaches zero, the secant of y = f(x) becomes a tangent to the y = f(x) curve. Thus, a s h tends to 0, the difference quotient yields the slope of the tangent and therefore the derivative of y = f. (x). i.e.,

Difference quotient calculate

The slope of the secant line passing through two points on the graph of f is computed using this formula. Also, these are the x-coordinate points x and x + h. The derivative is defined using the difference quotient. First, insert (x + h) wherever you see an x into your function.

Once you’ve determined f (x + h), plug your values into the difference quotient formula and simplify from there. We use the subtraction sign in the third step to remove the parentheses and simplify the difference quotient.

In addition to this, you will notice in the formal definition of the difference quotient that the slope we’re calculating is for the secant line. A secant line is simply any line that connects two points on a curve.

On our x-axis, we label these two points as x and (x +h). Because we are working with a function, these points are labelled on our y-axis as f (x) and f (x + h), respectively.

So, in layman’s terms, the difference quotient assists us in determining the slope of a curve. We cannot use the traditional formula of: in the case of a curve.

As a result, we must employ the difference quotient formula.

Difference quotient

Difference Quotient Formula

The following are the steps we take to determine the difference quotient:

  • Insert x + h into the function f and simplify to get f(x + h).
  • Now that you’ve calculated f(x + h), calculate f(x + h) – f(x) by plugging in f(x + h) and f(x) and simplifying.
  • Fill in your step 2 result for the numerator in the Newton quotient and simplify it.

Which, when taken to its logical conclusion as h approaches zero, yields the derivative of the function f The expression gets its name from the fact that it is the quotient of the function’s difference in values by the difference in the corresponding values of its argument (in this case, (x+h)-x=h).

Difference quotient

Symmetric difference quotient

In mathematics, a difference quotient is a formula that gives an approximation of a function’s derivative. There are two types of difference quotient: one-sided difference quotients and symmetric difference quotients. They are all related, and one provides a better approximation than the others because of this connection.

The symmetric derivative is a mathematical operation that generalises the ordinary derivative. It is defined as follows:

The expression below the limit is also known as the symmetric Newton quotient. If a function’s symmetric derivative exists at point x, it is said to be symmetrically differentiable at that point.

If a function is differentiable (in the usual sense) at a point, it is also symmetrically differentiable, but not vice versa.

Again, the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0, is a well-known counterexample.

Additionally, the symmetric Newton quotient provides a better numerical approximation of the derivative than the standard Newton quotient for differentiable functions.

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

For the symmetric derivative, neither Rolle’s theorem nor the mean value theorem hold; some similar but weaker statements have been proven.Read Also: Extemporaneous Speech – Definition, Examples

The modulus function

The modulus function graph. Take note of the sharp turn at x=0, which results in the curve being non-differentiable at x=0. As a result, the function has no ordinary derivative at x=0. However, the symmetric derivative exists for the function at x=0.

At x=0, we have the modulus function, f(x)=|x|.

 Since h>0, we have |h|=-(-h). As a result, we can see that the symmetric derivative of the modulus function exists and is equal to zero at x=0, despite the fact that its ordinary derivative does not exist at that point (due to a “sharp” turn in the curve at x=0).

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

How do you find the quotient?

To calculate the quotient, first, divide the dividend by the whole-number divisor.

Then, to make the divisor a whole number, multiply it by a power of ten.

Next, divide the dividend by the same power of ten. Insert the decimal point into the quote.

Then, to calculate the quotient, divide the dividend by the whole-number divisor.

What is quotient example?

The result of dividing one number by another is dividend/divisor = quotient.

Thus, for instance, in 12/3 = 4, 4 is the quotient division.

What is the formula of quotient?

The quotient rule is a formula for calculating the derivative of two functions that are quotients. If you have the function f(x) in the numerator and the function g(x) in the denominator, you can find the derivative using the following formula: The d in this formula denotes a derivative.

Difference quotient rational function

To find the Newton quotient of a rational function, you can follow these steps:

Firstly, identify f(x) and f(x+h). Also, here we will consider both the functions to be rational.

Secondly, insert the functions from the first step to the Newton quotient formula  [f(x + h) – f(x)] / h and then, simplify. Then, you will see that there is a common denominator between f(x) and f(x+h). Next, we will get the result after combining these rational functions, and transferring the denominator from our combined rational function to the denominator.

Difference quotient example

  • Find the difference quotient of the function f(x) = 4x – 7.

Solution:

Using formula,

Difference quotient of f(x) is given by :

= [ f(x + h) – f(x) ] / h

= [ (4(x + h) – 7) – (4x – 7) ] / h

= [ 4x + 4h – 7 – 4x + 7 ] / h

= [ 4h ] / h

= 4

Answer: The difference quotient of f(x) is 4.

  • Find the derivative of f(x) = 4x2 – 9 by applying the limit as h → 0 to the difference quotient formula.

Solution:

The difference quotient of f(x)

= [ f(x + h) – f(x) ] / h

= [ (4(x + h)2 – 9) – (4x2 – 9) ] / h

= [ (4 (x2 + 2xh + h2) – 9) – 4x2 + 9 ] / h

= [ 4x2 + 8xh + 4h2 – 4x2 ] / h

= [ 8xh + 4h2 ] / h

= [ h (8x + 4h) ] / h

= 8x + 4h

By applying the limit as h → 0, we get the derivative f ‘ (x).

f ‘(x) = 8x + 4(0) = 8x.

Answer: f ‘(x) = 8x.

  • Find the difference quotient of the function f(x) = ln x.

Solution:

Using the difference quotient formula, the difference quotient of f(x) is,

[ f(x + h) – f(x) ] / h

= [ ln (x + h) – ln x ] / h

=  ln [ (x + h) / x ] / h (because by quotient property of logarithms, ln m – ln n = ln (m / n))

Answer: The difference quotient of f(x) is, ln [ (x + h) / x ] / h.

Difference quotient calculator

The calculator computes the difference quotient of a function. The slope of a secant line drawn to a curve and passing through any two points on that curve is given by the Newton quotient formula. Thus, the Difference Quotient Calculator is an online tool for calculating the Newton quotient of a given function. So, the Newton quotient formula is part of the derivative definition. Enter the function in the provided input box to use this Newton quotient calculator.

Moreover, you can use the online difference quotient calculator to find the Newton quotient of the given function by following the steps outlined below:

  • In the first step: Navigate to any website’s Newton quotient calculator online.
  • Step 2: Enter the function into the Newton quotient calculator’s input box.
  • Then: To calculate the Newton quotient of the given function, click the “Calculate” button.
  • Step 4: Click the “Reset” button to clear the field and start over.

Difference quotient FAQs

1.   What is Difference quotient formula?

Ans. In very simple words, the slope of a secant line formula which we use in the very important expression [f(x + h) – f(x)] / h is the difference quotient of a function y = f(x).

2.   How to derive Difference quotient formula?

Ans. Because the very important expression is nothing more than the slope of a secant line, we derive the very important quotient formula from the slope formula. By slope formula, the slope of the line connecting (x, f(x)) and (x + h, f(x + h)) is [f(x + h) – f(x)] / [(x + h) – x] = [f(x + h) – f(x)] / h. This is the formula for the difference quotient.

3.   What Are the Applications of the Difference Quotient Formula?

Ans. This formula is most commonly used to calculate the derivative. In other words, the derivative of the function is given by the limit of the very important expression as h 0. In other words, f'(x) =

4.   How to Use the Difference Quotient Formula To Find the Derivative?

Ans. The limit of the very important expression of a function f(x) as h 0 is nothing more than the function’s derivative. In this case, the function is f(x). So, the derivative is: f'(x) =

5.   What is the application of difference quotient?

Ans. It has a wide range of real-world applications. In physics, we define instantaneous velocity of an object (the velocity at a specific point in time as the derivative of the object’s position (as function of time). For example, if an object’s position is given by x(t)=-16t^2+16t+32, its velocity is given by v(t)=-32t+16. Now, simply take the derivative of the instantaneous velocity function to find the instantaneous acceleration. In the preceding function, for example, the acceleration function is a(t) = -32.

6.   What does h mean in the formula of difference quotient?

Ans. in the expression, h represents the change in x or (x2 – x1) or (del)x.

On the other hand, f (x+h) – f (x)  represents (y2 – y1) or (del)y.

7.   What does the difference quotient tell you?

Ans. In simple words, the expression is a formula that finds the average rate of change of any function between two points