# Thread: Why getting different results when using product rule over quotient rule?

1. ## Why getting different results when using product rule over quotient rule?

Hello

I have a function:
$\displaystyle y = \frac{e^x}{x}$

If I differentiate using the quotient rule, then:
let u = e^x and let v = x

$\displaystyle \frac{dy}{dx} = \frac{x.e^x - e^x.1}{x^2}$

$\displaystyle = \frac{e^x(x - 1)}{x^2}$

But could I not express as a product ie $\displaystyle y = e^x . x^-1$

And then use the product rule,
$\displaystyle let u = e^x and v = x^-1$

$\displaystyle \frac{dy}{dx} = x^-1 . e^x - e^x . x^-2$

$\displaystyle \frac{e^x}{x} - \frac{e^x}{x^2} = \frac{e^(2x)}{x^2} - \frac{e^x}{x^2}$

$\displaystyle = \frac{e^(2x) - e^x}{x^2}$

Which is a different result! Why so? Or have I done it wrong?

2. ## Re: Why getting different results when using product rule over quotient rule?

You made a mistake on the second last line.

$\displaystyle \frac{e^x}{x} - \frac{e^x}{x^2} = \frac{xe^x}{x^2} - \frac{e^x}{x^2} = \frac{xe^x-e^x}{x^2}$

3. ## Re: Why getting different results when using product rule over quotient rule?

Originally Posted by SpringFan25
You made a mistake on the second last line.

$\displaystyle \frac{e^x}{x} - \frac{e^x}{x^2} = \frac{xe^x}{x^2} - \frac{e^x}{x^2} = \frac{xe^x-e^x}{x^2}$
Ah I squared top and bottom rather than multiply by x. I assumed squaring top and bottom would work.

I worked it out with a real example:

$\displaystyle \frac{3}{4} - \frac{1}{16}$

incorrect square method:
$\displaystyle \frac{9}{16} - \frac{1}{16} = \frac{8}{16} = \frac{1}{2}$ WRONG!

proper way - multiply top and bottom by 4
$\displaystyle \frac{12}{16} - \frac{1}{16} = \frac{11}{16}$ or (0.6875) CORRECT!

Well, I have learnt something. Thank you.

4. ## Re: Why getting different results when using product rule over quotient rule?

Consider the case of $\displaystyle \frac{4}{9}$. Is this equal to:

$\displaystyle \frac{4}{9}\times\frac{9}{9}$

...or $\displaystyle \frac{4}{9}\times\frac{4}{9}$?

You have to multiply the numerator and the denominator by the same value in order for them to be equivalent.

5. ## Re: Why getting different results when using product rule over quotient rule?

Originally Posted by Quacky
Consider the case of $\displaystyle \frac{4}{9}$. Is this equal to:

$\displaystyle \frac{4}{9}\times\frac{9}{9}$

...or $\displaystyle \frac{4}{9}\times\frac{4}{9}$?

You have to multiply the numerator and the denominator by the same value in order for them to be equivalent.
I answered my own question in the end. Yes agree (now) - thanks. I am learning by mistake

6. ## Re: Why getting different results when using product rule over quotient rule?

Originally Posted by angypangy
I answered my own question in the end. Yes agree (now) - thanks. I am learning by mistake
Mistakes are the gems of learning. Whenever you make a mistake, assuming the stakes are low such as in a classroom and not as in building a bridge, rejoice that you found it! Then hammer away at that mistake until you know exactly what you did wrong, and have come up with a method for avoiding that mistake in the future.