Implicit differentation problem

[physjeff12]

Homework Statement

find the lines that are (a)tangent and (b)normal to the curve at the given point:
x² - √(3)xy + 2y² = 5, (√3, 2)

Homework Equations

dy/dx

The Attempt at a Solution

i am completely confused about implicit differentiation and the chain rule. I've learned about 3 ways to do the chain rule and apparently they are all the same thing which confuses me more.

x² - √(3)xy + 2y² = 5
2y²(d/dx) = 5(d/dx) + √(3)xy(d/dx) - x²(d/dx)
4(dx/dy) = 0 + (1/2)3^-1/2y' - 2x
(dy/dy) = (((1/2)3^-1/2y' - 2x) / 4

and i am stuck.

 

[Mark44]

Homework Statement

find the lines that are (a)tangent and (b)normal to the curve at the given point:
x² - √(3)xy + 2y² = 5, (√3, 2)

Homework Equations

dy/dx

The Attempt at a Solution

i am completely confused about implicit differentiation and the chain rule. I've learned about 3 ways to do the chain rule and apparently they are all the same thing which confuses me more.

x² - √(3)xy + 2y² = 5
2y²(d/dx) = 5(d/dx) + √(3)xy(d/dx) - x²(d/dx)

d/dx as you are using it above doesn't mean anything. d/dx means "take the derivative with respect to x of whatever is to the right. dy/dx represents the derivative of y with respect to x. In implicit differentiation you are assuming that y is some differentiable function of x, and so are differentiating implicitly.

So let's start again from your second line, where you moved the terms around.
2y² = 5 + √(3)xy - x²
d/dx(2y²) = d/dx(5) + d/dx(√(3)xy) - d/dx(x²)
Two of the derivatives in the equation above are very straightforward, since you are differentiating a function of x, with respect to x. The other two require the chain rule. That is, the one on the left side and the 2nd one on the right side, which also requires the product rule (which has to be done first).

Can you carry out these differentiations?

4(dx/dy) = 0 + (1/2)3^-1/2y' - 2x
(dy/dy) = (((1/2)3^-1/2y' - 2x) / 4

and i am stuck.

 

[physjeff12]

oh so you can't take the derivative of 2y², the same as your take 2x²?

4y(y') = √3(√3)y - 2x
4y(y') = 3y - 2x

is that in the right direction?

 

[Mark44]

Partly. 4yy' is correct and -2x is correct, but the other one is not.
d/dx(√3xy) = √3 d/dx(xy) = ?
Hint: product rule

 

[physjeff12]

= √3(y+xy')
4yy' = √3y + √3xy'
y' = (√3y/4y) + (√3xy'/4y)
y' = (√3/4) + (√3xy'/4y)

closer?

 

[Mark44]

Closer, but still no cigar.
Start from here.
d/dx(2y²) = d/dx(5) + d/dx(√(3)xy) - d/dx(x²)

 

[physjeff12]

d/dx(2y²) = d/dx(5) + d/dx(√(3)xy) - d/dx(x²)
4yy' = √(3)y' - 2x
y' = (√(3)y')/4y - x/2y

any cigar yet?

 

[Mark44]

Nope. You didn't get the derivative right for d/dx(√(3)xy) this time. You had it in post 5, but apparently forgot that you needed the product rule.

 

[physjeff12]

oh i didn't know if i was doing it right.

d/dx(2y²) = d/dx(5) + d/dx(√(3)xy) - d/dx(x²)


4yy' = √3(y+xy') - 2x
y' = ((√3(y+xy')) - 2x)/4y

 

[Mark44]

That's correct, but you haven't solved for y' (aka dy/dx).
Start with this equation and get all the terms with y' on one side, and then divide both sides by whatever is the coefficient of y'.
4yy' = √3(y+xy') - 2x

 

[physjeff12]

4yy' = √3(y+xy') - 2x
4yy' + √3xy' = √3y - 2x
(4y + √3x)y' = √3y - 2x
y' = (√3y - 2x)/(4y + √3x)

right?

 

[Mark44]

No. What did you do to go from the first equation to the second?

 

[HallsofIvy]

Surely you do not believe that $\sqrt{(a+ b)}= \sqrt{a}+ \sqrt{b}$?

 

[Mark44]

Surely you do not believe that $\sqrt{(a+ b)}= \sqrt{a}+ \sqrt{b}$?

That's not applicable in this problem. physjeff12's expression is √3(y+xy'), which is the derivative of √3 xy. At least, this is how he presented it in his first post in this thread. I don't believe he intended the x and y factors to be in the radicand.

 

[Mark44]

4yy' = √3(y+xy') - 2x
4yy' + √3xy' = √3y - 2x

The second equation doesn't follow from the first. Do you know what you did wrong?