Derivatives - Quotient Rule

[Alpha6]

I was wondering if I did this right, and if not what to correct please.

I am looking for the derivative of z = 4-3x / 3x²

I understand the quotient rule as: Derivative of numerator * denominator Minus numerator times the Derivative of denominator all over Denominator squared.

So I did (4 -3X)' * (3X² + X) - (4-3X) * (3X²+ X)' / (3X²+X)²
↓

(3) * (3X² + X) - (4 - 3X) * (6X+1) / (3X²+X)²

↓

9X^{2 +}3X - 18X² -27X - 4 / (3X²+X)<sup>2

↓</sup>



-9X² -24X -4 / 9X⁴ +X²

 

[stapel]

I was wondering if I did this right, and if not what to correct please.

I am looking for the derivative of z = 4-3x / 3x²

What you have posted means this:

. . . . .\(\displaystyle z\, =\, 4\, -\, \dfrac{3x}{3x^2}\, =\, 4\, -\, \dfrac{1}{x}\)

However, I suspect that you meant "z = (4 - 3x)/(3x^2)", or:

. . . . .\(\displaystyle z\, =\, \dfrac{4\, -\, 3x}{3x^2}\)

Did I guess correctly?

So I did (4 -3X)' * (3X² + X) - (4-3X) * (3X²+ X)' / (3X²+X)²

How did the denominator, 3x^2, turn into 3x^2 + x? Where did the "+x" come from?

 

[HallsofIvy]

Assuming that the problem is to differentiate \(\displaystyle \frac{4- 3x}{3x^2}\) then, yes, you can use the quotient rule.

The quotient rule says that \(\displaystyle \left(\frac{u(x)}{v(x)}\right)'= \frac{u'(x)v(x)- u(x)v'(x)}{u^2(x)}\).

Here, that is \(\displaystyle \frac{(4- 3x)'(3x^2)- (4- 3x)(3x^2)'}{9x^4}\). What are \(\displaystyle (4- 3x)'\) and \(\displaystyle (3x^2)'\) ?
(In your first post, you have "3" as the derivative of 4- 3x. That is wrong.)

In this particular problem, since the denominator, \(\displaystyle 3x^2\) is a single term, you can write it as
\(\displaystyle (4/3)x^{-2}- x^{-1}\) and differentiate that. Do it both ways as a check.

 

[Alpha6]

mea culpa

I made a mistake posting the original equation, I should proofread it.

This is what it should be: z = (4-3x) / 3x² +x

That's where the "+x" came from sorry!

So I'll do it as...

(4-3x)' * (3x²+X) - (4-3x) * (3x² +x)'

↓

(-3) * (3x²+X) - (4-3x) * 6x

↓

-9x² -3x - ( 24x -18x²)

↓

-9x² -3x -24x + 18x²

↓
9x² - 27x