Implicit differentiation - Help

[pat666]

Homework Statement

Find d²y/dx² for cos y + x² =12
The first derivative I think is correct but the second I am unsure of, this is Q5 of 10 in an assignment and every other question has been far cleaner and easier making me think that I've missed something.
We don't have to use implicit differentiation, I thought that it would be easiest though.
thanks for any future help.

Homework Equations

The Attempt at a Solution

cos y=x-12
d/dx cos y = d/dx(12-x²)
dy/dx=-sin y = -2x
dy/dx=2x*cosec(x)
The first differential is easy, it gets messy from here.
d²y/dx²=d/dx*dy/dx
=d/dx 2x (sin y)^-1
=2/(sin y) +2x d/dx (sin y)^-1 -product
=2/(sin y) +2x [dy/dx -d/dy(sin y)^-1]
=2/sin y = (-4x² cos y)/(sin y)²
=2 cosec y + -4x² cot y cosec² y

 

[jegues]

dy/dx=-sin y = -2x

This line confuses me... Anyways let's give this a whirl!

$$\frac{d(cos(y) + x^{2}=12)}{dx}$$

$$2x - y'sin(y) = 0$$

EDIT: Whoops I made a mistake after this point, I'll try to post the correct solution in my next post.

 

[pat666]

thanks a lot jegues. Sorry if this is a bit rude but how sure are you that this is right? I only ask because the assignment is 20% of my final grade (ridiculous I know).

 

[hgfalling]

Wait on that chain rule step you have to use the product rule again: y' sin y isn't a function of sin y.

Anyway, OP, your answer is right, assuming you're allowed to express your answer implicitly. You can also set y = arccos(12-x²) and just differentiate directly, which after some mildly ugly algebra, results in an analytic expression in terms of x.

 

[pat666]

Anyway, OP, your answer is right, assuming you're allowed to express your answer implicitly. You can also set y = arccos(12-x²) and just differentiate directly, which after some mildly ugly algebra, results in an analytic expression in terms of x.

That was my first attempt but I have up when it got to ugly, also what is OP??

 

[hgfalling]

OP = "original poster", a common phrase on forums I've inhabited before, but not that common here now that I think about it.

 

[Char. Limit]

OP = Original Poster

And personally, I would take the derivatives implicitly. It's not that hard, and jegues was even nice enough to give you the first derivative.

However, just to let you know, posting the complete solution is against the PF rules. We want our friends to understand where the answer comes from, and such. I'm sure you understand.

From his first derivative, though, it should be easy to get the second.

 

[jegues]

OP, your answer is right

Sorry about the confusion, after going through my steps again I realized I had some mistakes. You are indeed right.

 

[pat666]

alright, now I am confused am I right or wrong?

 

[pat666]

thanks everyone - I am now assuming my OP was right.

 

[jegues]

alright, now I am confused am I right or wrong?

You are correct.

2 cosec y -4x2 cot y cosec2 y is the right answer.

Again, sorry for any confusion I may have caused.