Question

find dy/dx of sin(x)=x(1+tan(y))

Answer 1

@zepdrix last question of the night :)

Answer 2

Implicit? :o Oo neato

Answer 3

So whatchu think?
Set up that product rule, ya?

Answer 4

so cosx=tany+sec^2(dy/dx) ya?

Answer 5

\[\large\rm \sin(x)=x[1+\tan(y)]\]\[\large\rm \color{royalblue}{[\sin(x)]'}=\color{royalblue}{[x]'}[1+\tan(y)]+x\color{royalblue}{[1+\tan(y)]'}\]Hmm I feel like you're missing an x on the right side.
See my product rule set up?
See how I still have an x in one of the terms on the right?

Answer 6

Oh! I had it written down but forgot to type it here.

Answer 7

\[\large\rm \color{orangered}{\cos(x)}=\color{orangered}{[1]}[1+\tan(y)]+x\color{royalblue}{[1+\tan(y)]'}\]Orange = Differentiated.
Let's take it slow on this last guy

Answer 8

\[\large\rm \frac{d}{dx}(1+\tan y)=?\]

Answer 9

the 1 would become 0 and the tan y would be sec^2 (y) (dy/dx), is that right?

Answer 10

\[\large\rm \frac{d}{dx}(1+\tan y)=0+\sec^2(y)y'\]Ok great!

Answer 11

\[\large\rm \color{orangered}{\cos(x)}=\color{orangered}{[1]}[1+\tan(y)]+x\color{orangered}{[0+\sec^2(y)y']}\]So we've differentiated all of our stuff successfully.
Now we need to solve for y'.

Answer 12

\[\large\rm \cos x=1+\tan y+xy' \sec^2 y\]So how do we do that? :)
Any ideas?

Answer 13

you lost me. did you just move the y' to the x in front of sec^2(y)?

Answer 14

Yes :)
The y' is outside of the secant function.
So I just moved in front.
Too many brackets, was trying to clean things up a tad.

Answer 15

before you did that I could isolate y' and get (cosx-tany-1)/(x(sec^2)y) on the other side

Answer 16

We did a lot like that in class

Answer 17

Is there something wrong with that?

Answer 18

\[\large\rm \cos x=1+\tan y+xy' \sec^2 y\]Subtraction:\[\large\rm \cos x-1-\tan y=xy' \sec^2 y\]Division:\[\large\rm \frac{\cos x-1-\tan y}{x \sec^2y}=y'\]Perfect! \c:/

Answer 19

I'm not sure what (sec^2)y is.
Careful with the way you do your brackets :)

Answer 20

written down it looks identical to yours

Answer 21

i'm just not too good at getting this stuff typed

Answer 22

oh i see :D

Answer 23

Until you become a \(\large \color{orange}{master}\) at these cool little rules,
I would strongly recommend you `set up your product rules before doing them`.

I should take that back actually..
If you're the type of student who is prone to making lots of little errors,
then maybe the less you write down the better.
I've known quite a few students who are like that.

But personally I really like the set up.

Answer 24

Refer to the attachment using Mathematica v9