# My Puzzling Work on Differentiating a Fraction

• snowJT
In summary, the person found the IC of d(\frac{x^2}{y}) by solving for x and y using the quotient rule and then using the product rule.
snowJT
Its funny.. because I got this right on a test.. but.. I'm looking back at it.. and it doesn't make sense how I figured it out...

I had to find the IC of $$d(\frac{x^2}{y})$$

this was my work...

$$d(\frac{x^2}{y}) = 2xy - x^2y^2 - y^2$$
$$d(\frac{x^2}{y}) = \frac{2xy - x^2y}{y^2}$$

I'm just confused because I would of thought that it would of been something like this...

$$d(\frac{x^2}{y}) = 2xy^-^1 + x^2y^-^2$$

hmm...?

snowJT said:
Its funny.. because I got this right on a test.. but.. I'm looking back at it.. and it doesn't make sense how I figured it out...

I had to find the IC of $$d(\frac{x^2}{y})$$

this was my work...

$$d(\frac{x^2}{y}) = 2xy - x^2y^2 - y^2$$
$$d(\frac{x^2}{y}) = \frac{2xy - x^2y}{y^2}$$

I'm just confused because I would of thought that it would of been something like this...

$$d(\frac{x^2}{y}) = 2xy^-^1 + x^2y^-^2$$

hmm...?
What do you mean by "IC"? When I saw "Solving an IC" in the differential equation forum, I thought you mean an "initial condition" problem but I see no differential equation at all here!

Certainly, the derivative of x2/y is, by the quotient rule,
$$\frac{2xy- x^2}{y^2}$$
There is no "y" in the second term in the numerator. (I surely don't see where you get your first line
$$d(\frac{x^2}{y}) = 2xy - x^2y^2 - y^2$$
that makes no sense to me.)

Of course, you could also write x2/y as x2y-1 and use the product rule. In that case, the dervative is
$$2xy^{-1}+ (-1)x^2y^{-2}$$
Just rewrite that as
$$\frac{2x}{y}- \frac{x^2}{y^2}$$
multiply the numerator and denominator of the first fraction by y and combine and you get the same answer as with the quotient rule.

Since this has nothing to do with differential equations, I moving it to calculus.

sorry, I didn't know the difference, this was on a DE test

but thanks for explaining it

## 1. What is a fraction?

A fraction is a number that represents a part of a whole. It is written as a ratio of two numbers, the top number being the numerator and the bottom number being the denominator.

## 2. How do you differentiate a fraction?

To differentiate a fraction, you must find the derivative of the numerator and denominator separately and then divide the two derivatives. The quotient rule is often used for this process.

## 3. Why is differentiating a fraction important?

Differentiating a fraction is important because it allows us to find the rate of change of a quantity. This is useful in many applications, such as finding the velocity of an object or the growth rate of a population.

## 4. Can fractions with variables be differentiated?

Yes, fractions with variables can be differentiated. The process is the same as differentiating fractions with numbers, except the derivative will also contain the variable.

## 5. Are there any special cases when differentiating a fraction?

Yes, there are some special cases when differentiating a fraction. These include fractions with radicals, fractions with trigonometric functions, and fractions with logarithms. In these cases, special rules or techniques may be needed to find the derivative.

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