# Question on derivative question involving quotient & chain rule

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• Oct 19th 2009, 02:53 PM
Archduke01
Question on derivative question involving quotient & chain rule
There are 2 questions involving derivatives that I have trouble getting to the answer for. The first one is \$\displaystyle y = [(6-5x) / (x^2 - 1)]^2\$

and the 2nd one is

\$\displaystyle y = 36 / (3-x)^2\$

The answers respectively are;
\$\displaystyle [2(6 - 5x)(5x^2 - 12x + 5)] / (x^2 - 1)^3\$
&
\$\displaystyle y = (8/3)x + 4\$

It would be very much appreciated if anyone could provide step-by-step solutions to both problems.
• Oct 19th 2009, 03:07 PM
tom@ballooncalculus
Just in case a picture helps...

http://www.ballooncalculus.org/asy/d...d/quadBott.png

... where

http://www.ballooncalculus.org/asy/chain.png

... is the chain rule, and...

http://www.ballooncalculus.org/asy/prod.png

... the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which subject to the chain rule).

So what you've got here is two chains inside a product-rule (the legs-uncrossed version). That's just my preference - you could do it as a product (or even a quotient - yeughh!) inside a chain.

Anyway, if you feel up to filling in the blanks and simplifying, notice I've tweaked the third balloon along for the sake of the common denominator.

Hope that helps, or doesn't further confuse.

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• Oct 19th 2009, 03:28 PM
Archduke01
Do you have a solution for the 2nd problem by any chance?
• Oct 19th 2009, 03:32 PM
tom@ballooncalculus
Hate to say this (after last time) but are you QUITE sure that's the right derivative?
• Oct 19th 2009, 03:33 PM
Archduke01
For the 2nd question?
• Oct 19th 2009, 03:35 PM
tom@ballooncalculus
Oh yes
• Oct 19th 2009, 03:39 PM
Archduke01
Oh damn, the answer must have confused you because it's not the answer to the derivative directly, but an equation where the point is (0,4).

Because the question is to find an equation of the tangent line to the graph of the function at the given point. I'm terribly sorry for overlooking that!
• Oct 19th 2009, 03:49 PM
tom@ballooncalculus
Oh, fine. The derivative is the first step, it's not a hard one, just the chain rule in a simple way. I'll post a pic just in case it helps, but go ahead and try it, then I guess you know to plug in zero as the x-value in the derivative function, to find the slope of the tangent line...

http://www.ballooncalculus.org/asy/diffProd/frac.png

No need to simplify or anything, just plug zero (as the value of x) into the derivative and you have the slope of the line. Now, the y-value where x is zero has a special role in one form of the line's equation, no?
• Oct 19th 2009, 03:58 PM
Archduke01
Oh yeah, the second part is easy. It's only the derivative part that I'm concerned about.

Could you please include all the steps for the first question? I don't know how they got to that long answer, mine doesn't match us.
• Oct 19th 2009, 04:01 PM
tom@ballooncalculus
Well, can you see that the denominators match?
• Oct 19th 2009, 04:04 PM
tom@ballooncalculus
Put it another way... can you see why the final (correct) denominator is (x^2 - 1)^3
?
• Oct 19th 2009, 04:06 PM
Archduke01
My denominator matches that of the answer, but I must've messed up somewhere on the calculations of the numerator because they don't match.
• Oct 19th 2009, 04:09 PM
tom@ballooncalculus
OK, you're ahead of me, so I don't know for sure if the numerator will match*, but do you get (for the numerator) the following...

\$\displaystyle 2(6 - 5x)(-5)(x^2 - 1) + (6 - 5x)^2 (-2) (2x)\$

If not, read off from the bottom row of balloons

*Yep, it does
• Oct 19th 2009, 04:13 PM
Archduke01
Quote:

Originally Posted by tom@ballooncalculus
OK, you're ahead of me, so I don't know for sure if the numerator will match, but do get (for the numerator) the following...

\$\displaystyle 2(6 - 5x)(-5)(x^2 - 1) + (6 - 5x)^2 (-2) (2x)\$

If not, read off from the bottom row of balloons

I got all of the above values except the \$\displaystyle (x^2 - 1) + (6 - 5x)^2\$. How did you get these two?
• Oct 19th 2009, 04:17 PM
tom@ballooncalculus
Well, I can only suggest checking yours against the bottom row of balloons and checking that the balloons are following the product-rule, as they should. The product rule may be hard to see amongst all the chain rule details so I might simplify...

By the way, beware the 'tweak', third balloon along, which is for the sake of the common denominator.
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