C3 Diff. Products - I'm so confused as to what I am doing wrong.

From my understanding your supposed to use the quotient rule which I have been doing and my final answer is always coming out to be 8y = 5x + 3

however in the answers it says the answer is 4y = x + 3

Can somebody do a quick step by step please? Many thanks and i'll give rep+ and all that good stuff.

Textbook: Cambridge Advanced Mathematics Core 3 & 4 by Hugh Neil and Douglas Quadling

Chapter 10 Diff Products: Ex 10B page 171.

Question 5.

Find the equation of the tangent at the point with coordinates (1,1) to the curve with equation y= (x^2 +3)/(x + 3)


From my understanding your supposed to use the quotient rule which I have been doing and my final answer is always coming out to be 8y = 5x + 3

however in the answers it says the answer is 4y = x + 3

Can somebody do a quick step by step please? Many thanks and i'll give rep+ and all that good stuff.

(If you have gone wrong that is. I haven't checked.)

(2x)(x+3) - (x^2 + 3)(1) / (x+3)^2 = (x^2 + 6x + 3) / (x+3)^2 = gradient of tangent

You sure about that?

yeah i am sure it is +3. isn't it?

(2x)(x+3) - (x^2 + 3)(1) = (2x^2 + 6x) - (x^2 + 3) = x^2 + 6x + 3

-(+3)=+3?

lol. im blind. i'll try that and see if get the right answer.

Sir, you have the eye of an eagle. Thank you. It turns out that's what I was doing wrong. Thank you very much.

I don't really get all the negative rep on my comment. I seriously think you should consider throwing the quotient rule out the window. You might forget it, or just get it confused. The key to doing well is keeping it simple, and not relying on memorised formulas without understanding what is going on.

Judging by the timestamp on this post and the neg-rep I just received I'm guessing that was from you... but for what it's worth, I never negged you. In fact, I agree with your sentiment. I think the quotient rule and quadratic formula are worth memorising though, provided you know how to derive them too, because they're very useful.

"Providing you know how to derive them" is the key!!

Well sort of. You can use the chain and product rules to derive the quotient rule by writing $\dfrac{f(x)}{g(x)} = f(x)g(x)^{-1}$, but can you derive the product rule? Can you derive the chain rule? In fact, before even proceeding you should be able to check if $f,g$ are both differentiable and we should calculate their derivatives from first principles. I mean, why would you just quote the result $\dfrac{d}{dx} (x^n) = nx^{n-1}$ when you can derive it from first principles using the binomial theorem? After all, it's very easy to screw up if you're just memorising formulae. But whilst we're at it, we should really derive the binomial theorem. I suppose we can do this by induction, but it seems silly to use induction if we haven't defined what the integers are, so we'd better get Peano on this question's ass.

Sarcasm aside, it's very difficult to know where to draw the line with things like this, so it seems silly to say that you should never memorise certain things unless you know how to derive them.