How do i differentiate y=x-2/(x+3)^2 and y=(x+1) sqrt(x-1)

The first i've tried using the chain rule then the quotient rule but cannot get the right answer can anyone help and show the working

By the way, a serious answer to the first using the chain rule:

\frac{x-2}{(x+3)^2} = \frac{u-5}{u^2}

where

u=x+3

. That is

\frac{1}{u} - \frac{5}{u^2}

, which we differentiate wrt u to get

\frac{-1}{u^2} + \frac{10}{u^3}

. Note that

u = x +3 \implies \dfrac{du}{dx} = 1

, so "differentiation wrt u" is the same as "differentiation wrt x", by the chain rule. Re-substitute, to get

\frac{-1}{(x+3)^2} + \frac{10}{(x+3)^3}

.

I wouldn't use this method unless I could spot a linear substitution (in this instance,

u = x+3

), though, because pretty much any other substitution makes it suddenly very easy to get confused (since we no longer have "differentation wrt u = differentiation wrt x").

Reply 23

theDanIdentity

16

Original post by Rickstahhh

You can't use product rule on this, as there is not two functions of x multiplied together.

Imo quotient rule is the easiest option

actually...

..you could...

its' just that its' long and complicated and not advised by me (and the C3 text book lool). but it is possible to use the product rule for quotient questions; but i don't know about using the quotient rule for product questions tho.. if any one thinks this is possible.. please quote me and reply asap

Reply 24

Rickstahhh

6

Original post by theDanIdentity

actually...

..you could...

its' just that its' long and complicated and not advised by me (and the C3 text book lool). but it is possible to use the product rule for quotient questions; but i don't know about using the quotient rule for product questions tho.. if any one thinks this is possible.. please quote me and reply asap

You can use quotient rule for functions of x multiplied together, but only if one of them has a negative power. Which would then theoretically mean that one is divided by the other.
I think that is the only situation when you can?

Reply 25

Smaug123

15

Original post by theDanIdentity

but it is possible to use the product rule for quotient questions; but i don't know about using the quotient rule for product questions tho.. if any one thinks this is possible.. please quote me and reply asap

f(x) g(x) = \frac{f(x)}{g(x)^{-1}}

. Quotient rule on that.

Reply 26

theDanIdentity

16

Smaug.. Youre. A. ****ing. Genius. Man.

Woww.. You double the negative ..?
So the 'g' would be to the negative twice as its on the bottom WITH a negative power? Am I right or just plain wrong?
Does.. Does it work ?!

Reply 27

Mathlover123

11

Original post by theDanIdentity

actually...

..you could...

its' just that its' long and complicated and not advised by me (and the C3 text book lool). but it is possible to use the product rule for quotient questions; but i don't know about using the quotient rule for product questions tho.. if any one thinks this is possible.. please quote me and reply asap

In honesty it tends to make it easier, this is a university Maths student saying this btw. But quotient rule will take much longer than just using the product rule with a negative power in like 75% of cases.

Reply 28

Smaug123

15

Original post by theDanIdentity

Smaug.. Youre. A. ****ing. Genius. Man.

Why thank you. I've often thought so.

Woww.. You double the negative ..?
So the 'g' would be to the negative twice as its on the bottom WITH a negative power? Am I right or just plain wrong?
Does.. Does it work ?!

It does work. It's very rarely useful, because the product rule is just quite a lot easier than the quotient rule, almost always. (Even on quotients.)

Mei C3 differentiation help

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