differentiation help

[megynbrown1]

ANYONE OUT THERE! I AM STUCK. I have these 4 differentiation problems that are killing me. Can anyone help me with any of them. I am willing to do the work, I just need guidance! - Megs

1.) Find dy/dx by implicit differentiation.

x^3 + y^3 = 5

I think it is x^2 / y^2 but I am not exact!

2.) 2xy - y^2 = 1

I think this one is y/ x-y

3.) (y^3)(e^x) + x = y^6(x)

I don't even know how to start this one

4.) x ln y + y = (x^4)(y^4)

I don't even know how to start this one

Thanks to who ever has the patience to help me!

 

[pka]

In each of these think of y as a function of x.
\(\displaystyle \L
x^3 + y^3 = 5\quad \Rightarrow \quad 3x^2 + 3y^2 \frac{{dy}}{{dx}}=0\)

 

[megynbrown1]

are my 1 and 2 right?

3 ???

and 4 ???

 

[galactus]

How about if I do #3. You can tackle the rest. Implicit differentiation isn't that bad once you get the hang of it.

Think of the product rule. You know it, I assume?.

We have:

\(\displaystyle \L\\y^{3}e^{x}+x=xy^{6}\)

Now, let's do the \(\displaystyle y^{3}e^{x}+x\)

First times derivative of second plus second times derivative of first.

When you differentiate y, tack on a dy/dx. Here we go:

\(\displaystyle y^{3}\) is the first and \(\displaystyle e^{x}\) is the second. See?.

\(\displaystyle y^{3}e^{x}+e^{x}3y^{2}\frac{dy}{dx}+1\)

Now, do the other part:

\(\displaystyle xy^{6}\)

\(\displaystyle \L\\x6y^{5}\frac{dy}{dx}+y^{6}(1)\)

Put it all together:

\(\displaystyle \L\\y^{3}e^{x}+e^{x}3y^{2}\frac{dy}{dx}+1=6xy^{5}\frac{dy}{dx}+y^{6}\)

Bring everything to the left side and factor out dy/dx:

\(\displaystyle \L\\\frac{dy}{dx}(e^{x}3y^{2}-6xy^{5})+e^{x}y^{3}+1-y^{6}=0\)

Solve for dy/dx:

\(\displaystyle \L\\\frac{dy}{dx}=\frac{-1+y^{6}-e^{x}y^{3}}{3e^{x}y^{2}-6xy^{5}}\)

Do some tidying up:

\(\displaystyle \L\\\frac{dy}{dx}=\frac{-(e^{x}y^{3}-y^{6}+1)}{3y^{2}(e^{x}-6xy^{5}})\)

You can bring the right side to the left side first, if you like, and then differentiate. I just thought I would baby-step it for you to help out.

See, now?. Try the others. See what you get and write back with your attempts. Someone can check your answers.