Question

Find the differential dy of the given function. (Use "dx" for dx).

y = x+1
-----
5x+9

Answer 1

Haha cool name/ picture XD

Answer 2

thanks

Answer 3

\[\Large\sf y=\frac{x+1}{5x+9}\]Looks like we need to apply the quotient rule, yes?

Answer 4

yes

Answer 5

So here's our quotient rule setup:
\[\Large\sf \color{royalblue}{dy}=\frac{\color{royalblue}{d(x+1)}(5x+9)-(x+1)\color{royalblue}{d(5x+9)}}{(5x+9)^2}\]
Hopefully my notation is ok for differentials.
So we need to take the derivative of the blue parts, (left side is already in the form we need).

Answer 6

ok.
(1)(5x+9)-(x+1)(5)
----------------
(5x+9)^2

Answer 7

Ok good, let's remember to keep our differentials for x though,\[\Large\sf \color{royalblue}{dy}=\frac{\color{royalblue}{dx}(5x+9)-(x+1)\color{royalblue}{5dx}}{(5x+9)^2}\]

Answer 8

Then simply "factor" dx out of each term,\[\Large\sf \color{royalblue}{dy}=\frac{(5x+9)-5(x+1)}{(5x+9)^2}\color{royalblue}{dx}\]And simplify! :)

Answer 9

You would already have your form for the differential dy. Generally there is no need to simplify beyond having your derivatives evaluated.

Answer 10

I don't know how else to simplify it

Answer 11

If you wanted to simplify the algebra work, you could use distributive property on the -5(x + 1). Then combine like terms in the numerator. It isn't quite necessary if your goal was only to find dy, although it makes the differential easier to work with that way.

Answer 12

\[\frac{ 4 }{(5x+9)^{2}}\]

Answer 13

Looks good to me. :)

Answer 14

yay good job \c:/