How are these pattered in the rule of derivatives?
Find the derivative of dy/dx of the relations:
1.) x^3y+ 2y^4 - x^4 = 2xy
and
2.) x^2 = x+2y/ x - 2y
thanks a lot
If I want to simplify $\displaystyle \frac{d(y)}{dx}$, I obviously can't do it that easily as the variables are wrong.
However, I can apply the chain rule: $\displaystyle \frac{d(y)}{dx}.\frac{dx}{dy}=\frac{d(y)}{dy}$
Solving for $\displaystyle \frac{d(y)}{dx}$, I get $\displaystyle \frac{d(y)}{dx}=\frac{d(y)}{dy}.\frac{dy}{dx}$
Evaluate $\displaystyle \frac{d(y)}{dy}$ which is $\displaystyle 1$. Therefore $\displaystyle \frac{d(y)}{dx}= \frac{dy}{dx}$
Now try apply these principles in your question
To be honest, I don't understand this. What is $\displaystyle \frac{d(y)}{dx}$ and how is it different from $\displaystyle \frac{dy}{dx}$?
OP, it seems that you need to differentiate an implicit function. Obviously, you need to read how to do this, for example, in Wikipedia. Once you learn the method and read several finished examples, please describe the exact difficulty you are having with this particular problem. The thing is that this forum is not a tutorial service and you can't expect people to explain completely new material to you. The forum is here to help those who know the material but struggle to apply it in a concrete situation.
i'll give you an example, perhaps you'll see how it works:
if x^{2}+y^{2} = 1, find dy/dx.
step 1) differentiate both sides with respect to x:
2x(dx/dx) + 2y(dy/dx) = 0 but dx/dx = 1, so:
2x + 2y(dy/dx) = 0
step 2) solve for dy/dx:
2y(dy/dx) = -2x
dy/dx = -2x/2y = -x/y
So this is called implicit differentiation. You can differentiate both sides with respect to $\displaystyle x$ where $\displaystyle y$ is a function of $\displaystyle x$ and $\displaystyle \frac{dx}{dx} = 1$:
$\displaystyle \begin{aligned} x^3 y + 2 y^4 -x^4 & = 2xy \\ x^3 \frac{dy}{dx} + 3x^2 y + 8y^3 \frac{dy}{dx} - 4x^3 & = 2y + 2x \frac{dy}{dx} \\ \\ [\text{ Now you gather dy/dx one side and isolating dy/dx }] \\ \\ \frac{dy}{dx} & = \frac{2y - 3x^2 y + 4 x^3}{x^3 + 8 y^3 - 2x} \end{aligned} $
For the second one:
$\displaystyle \begin{aligned} x^2 & = x + \frac{2y}{x} - 2y \\ 2x & = 1 + \frac{x \frac{2 dy}{dx} - 2y}{x^2} - 2 \frac{dy}{dx} \\ \\ [\text{Same as before gather dy/dx together and simplifying using Algebra you get }] \\ \\ \frac{dy}{dx} & = \frac{2x^3 -x^2 + 2y}{2x - 2x^2} \end{array} $
You can check it to confirm here for the first equation:
differentiation x^3 y +2 y^4 -x^4 - 2xy = 0 - Wolfram|Alpha
And for the second one:
differentiation x^2 - x - 2y/x +2y = 0 - Wolfram|Alpha
Hope this helps.
x3bnm read the question AS YOU WROTE IT. If you had used the brackets in the correct place in the first place, you would have been given a correct solution.
Anyway, you now know how to differentiate implicitly, so I suggest you try it out for yourself with the correct question.
Yes the new solution is given below:
$\displaystyle \begin{align*} x^2 =& \frac{(x+2y)}{(x-2y)} \\ 2x =& \frac{(x - 2y)(1 + 2 \frac{dy}{dx}) - (x+2y)(1 - 2 \frac{dy}{dx})}{(x-2y)^2} \\ 2x =& \frac{x + 2x\frac{dy}{dx} - 2y -4y \frac{dy}{dx} -x + 2x\frac{dy}{dx} -2y +4y\frac{dy}{dx}}{(x -2y)^2} \\ 2x(x - 2y)^2 =& x + 2x\frac{dy}{dx} - 2y -4y \frac{dy}{dx} -x + 2x\frac{dy}{dx} -2y +4y\frac{dy}{dx} \\ 2x(x - 2y)^2 + 4y =& 4x \frac{dy}{dx} \\ \frac{dy}{dx} =& \frac{2x(x - 2y)^2 + 4y}{4x} \\ \frac{dy}{dx} =& \frac{x(x-2y)^2 + 2y}{2x} \\ \frac{dy}{dx} =& \frac{x^3 - 4x^2 y + 4 xy^2 + 2y}{2x} \text{.................[after simplification] } \end{align*}$
Hope it will help you.
Also you can check the answer at:
differentiation x^2 = (x+2y)/(x-2y) - Wolfram|Alpha