Implicit differentiation find points where slope is ___

I have two problems dealing with this. I am getting myself all screwed up and could use help.

the first is at what points is the slope of the tangent line equal to 1 on [x^2/9] + [y^2/16] = 1

So I got the derivative dy/dx = -[16x/9y]

so now I solve for -[16x/9y] = 1 ; correct?

The other question is find the points at which the tangent is horizontal for

x^3 + 4x^2 +xy^2 - 4y^2 = 0

I got dy/dx = [3x^2 + 8x +y^2]/6

so same thing, solve for dy/dx to be zero?

Re: Implicit differentiation find points where slope is ___

Quote:

Originally Posted by

**Jonroberts74** I have two problems dealing with this. I am getting myself all screwed up and could use help.

the first is at what points is the slope of the tangent line equal to 1 on [x^2/9] + [y^2/16] = 1

So I got the derivative dy/dx = -[16x/9y]

so now I solve for -[16x/9y] = 1 ; correct?

Yes, when you have $\displaystyle -\dfrac{16x}{9y} = 1$, you solve for one of the variables (either x or y) and plug it back into the original equation for the eclipse: e.g. $\displaystyle y = -\dfrac{16x}{9}$, so $\displaystyle \dfrac{x^2}{9} + \dfrac{\left(-\dfrac{16x}{9} \right)^2}{16} = 1$.

Quote:

Originally Posted by

**Jonroberts74** The other question is find the points at which the tangent is horizontal for

x^3 + 4x^2 +xy^2 - 4y^2 = 0

I got dy/dx = [3x^2 + 8x +y^2]/6

so same thing, solve for dy/dx to be zero?

Yes, you do the same thing to get a relationship between x and y. But you may want to check what you got for the derivative once more. You should have gotten $\displaystyle \dfrac{dy}{dx} = \dfrac{3x^2 + 8x + y^2}{8y- 2xy}$.

Re: Implicit differentiation find points where slope is ___

ah yeah, oops. got the correct derivative the second time around.

Re: Implicit differentiation find points where slope is ___

Quote:

Originally Posted by

**Jonroberts74** ah yeah, oops. got the correct derivative the second time around.

Also for that second problem, when you plug in zero for the derivative, you can solve for $\displaystyle y^2$ since the original formula does not have any terms with only $\displaystyle y$ in them. This way, you avoid square roots.

Re: Implicit differentiation find points where slope is ___

Can you show me how you would solve to get that, I'm not getting it. And I tried using wolfram but it's losing me. It's saying multiple both sides by the denominator to clear the fraction. Which I understand in other problems fine but it's confusing me with this.

thanks.

Re: Implicit differentiation find points where slope is ___

$\displaystyle \dfrac{dy}{dx} = \dfrac{3x^2 + 8x + y^2}{8y-2xy}$

Set the derivative equal to zero and solve for $\displaystyle y^2$:

$\displaystyle 0 = \dfrac{3x^2 + 8x + y^2}{8y-2xy}$

$\displaystyle 0 = 0\cdot(8y-2xy) = \dfrac{3x^2 + 8x + y^2}{8y-2xy}(8y-2xy) = 3x^2+8x+y^2$

$\displaystyle y^2 = -3x^2-8x$

Plug this into the original equation $\displaystyle x^3 + 4x^2 + xy^2 -4y^2=0$:

$\displaystyle \begin{align*}x^3 + 4x^2 + x(-3x^2-8x) -4(-3x^2-8x) & = 0 \\ x^3 + 4x^2 -3x^3 - 8x^2 + 12x^2 + 32x & = 0 \\ -2x^3+8x^2+32x & = 0 \\ -2x(x^2-4x-16) & = 0 \end{align*}$

Re: Implicit differentiation find points where slope is ___

okay perfect, then input the value of x into the original to find the y point. correct?

Re: Implicit differentiation find points where slope is ___

Or plug them into $\displaystyle y^2 = -3x^2-8x$

Re: Implicit differentiation find points where slope is ___

I was just going to ask that haha

Re: Implicit differentiation find points where slope is ___

should have mentioned there was a diagram showing that the x value wouldn't be exceeding 4 for the the right value for x is x = 2 - 2sqrt5

so plugging that in I get

y^2 = -3[2-2sqrt5] - 8[2 - 2sqrt5]

distribute those out for y = sqrt[-22 + 22sqrt5] but thats not the right answer. what am I missing?

Re: Implicit differentiation find points where slope is ___

the answer comes out to (2 - 2sqrt5, 2 +/- sqrt[10sqrt(5) - 22])

Re: Implicit differentiation find points where slope is ___

oh I missed a square. try this now

Re: Implicit differentiation find points where slope is ___

nope, still not getting it.

Re: Implicit differentiation find points where slope is ___

I get to -56 +40sqrt5 = y^2 from in the value of x into y^2 = -3x^2 -8x

Re: Implicit differentiation find points where slope is ___

solved, got it, thank you