# Differentiation of a Quotient check

• DBeckett91
You did it correctly.In summary, the provided equation involves finding the derivative of (8x^2 - 7x + 125)/(8x+7) where the value of x is unknown. Using the quotient rule, the derivative is calculated to be (64x^2+951)/(8x+7)^2.
DBeckett91

## Homework Statement

S = (8x^2 - 7x + 125)/(8x+7)

Where the value of x is unknown

## The Attempt at a Solution

u = 8x^2 - 7x + 125 du/dx = 16x - 7

v = 8x + 7 dv/dx = 8

ds/dx = (v*du/dx - u*dv/dx)/v^2

= (8x+7*16x-7 - 8x^2-7x+125*8)/(8x+7)^2

= (128x^2 - 56x + 112x - 49) - (64x^2 - 56x + 1000)/(8x+7)^2

= (128x^2 + 56x - 49) - (64x^2 - 56x + 1000)/(8x+7)^2

= (64x^2+951)/(8x+7)^2

This rule I'm not very comfortable or confident with so if I have gone wrong anywhere in my equations any help would be much appreciated

DBeckett91 said:

## Homework Statement

S = (8x^2 - 7x + 125)/(8x+7)

Where the value of x is unknown

## The Attempt at a Solution

u = 8x^2 - 7x + 125 du/dx = 16x - 7

v = 8x + 7 dv/dx = 8

ds/dx = (v*du/dx - u*dv/dx)/v^2

= (8x+7*16x-7 - 8x^2-7x+125*8)/(8x+7)^2
You would do better to use parentheses:
[(8x+7)(16x- 7)- (8x^2- 7+ 125)(8)]/(8x+ 7)^2

= (128x^2 - 56x + 112x - 49) - (64x^2 - 56x + 1000)/(8x+7)^2
and here you mean [(128x^2 - 56x + 112x - 49) - (64x^2 - 56x + 1000)]/(8x+7)^2

= (128x^2 + 56x - 49) - (64x^2 - 56x + 1000)/(8x+7)^2
The only difference between this and the previous line is that the "112x" is missing. What happened to it?

= (64x^2+951)/(8x+7)^2

This rule I'm not very comfortable or confident with so if I have gone wrong anywhere in my equations any help would be much appreciated

HallsofIvy said:
You would do better to use parentheses:
[(8x+7)(16x- 7)- (8x^2- 7+ 125)(8)]/(8x+ 7)^2

and here you mean [(128x^2 - 56x + 112x - 49) - (64x^2 - 56x + 1000)]/(8x+7)^2

The only difference between this and the previous line is that the "112x" is missing. What happened to it?

In the above line there is -56x and +112x so all I did was 112x-56x to gain +56x, it was just some simplification of the equation so I had fewer numbers to work with later on

Oh, I see. Yes, that is the correct answer.

## What is the quotient rule for differentiation?

The quotient rule for differentiation is a rule that allows us to find the derivative of a quotient of two functions. It states that the derivative of a quotient is equal to the denominator multiplied by the derivative of the numerator, minus the numerator multiplied by the derivative of the denominator, all divided by the square of the denominator.

## How do I apply the quotient rule to differentiate a function?

To apply the quotient rule, we first identify the numerator and denominator of the quotient. Then, we find the derivative of each individual function using the power rule, product rule, or chain rule as needed. Finally, we plug these derivatives into the quotient rule equation to find the derivative of the quotient.

## Can the quotient rule be used for all types of functions?

Yes, the quotient rule can be used for any type of function, as long as it is a quotient of two functions. However, it is important to note that the quotient rule can be more complicated and time-consuming than other differentiation rules, so it may not always be the most efficient method.

## What are the common mistakes to avoid when using the quotient rule?

One common mistake when using the quotient rule is forgetting to square the denominator in the final step of the equation. Another mistake is incorrectly applying the product rule when finding the derivative of the numerator or denominator. It is also important to be careful with parentheses and signs when plugging in the derivatives to the quotient rule equation.

## Are there any alternative methods for differentiating a quotient?

Yes, there are alternative methods for differentiating a quotient. One method is to rewrite the quotient as a product and use the product rule. Another method is to use the chain rule by treating the quotient as a composite function. However, it is important to compare the efficiency and accuracy of these methods with the quotient rule before choosing the best approach.

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