find the derivative of ln(x^2)-3y=2-3x(lny)

I know this is an implicit differentiation, and have gotten as far as:

dy/dx=((lny)(-3)-(2x/x^2))/(-3+(3x/y)) My friend and I have been working on it for over an hour. If someone could please give a step by step answer, we would appreciate it so much!!

Question

find the derivative of ln(x^2)-3y=2-3x(lny)

I know this is an implicit differentiation, and have gotten as far as:

dy/dx=((lny)(-3)-(2x/x^2))/(-3+(3x/y)) My friend and I have been working on it for over an hour. If someone could please give a step by step answer, we would appreciate it so much!!

myininaya · Answer

let's do this term by term
$$\frac{d}{dx}\ln(x^2)=2 \frac{d}{dx} \ln(x)=?$$

anonymous · Answer

Here is a more clear representation of what we have

$$\frac{ dy }{ dx }= \frac{ (lny)(-3)-(\frac{ 2x }{ x^2 }) }{ -3+(\frac{ 3x }{ y }) }$$

anonymous · Answer

$$\frac{ dy }{ dx }(lnx^2)=\frac{ 1 }{ x }(2x)$$

myininaya · Answer

well actually it would be 2x/x^2

myininaya · Answer

or just 2/x

myininaya · Answer

derivative of inside/inside

anonymous · Answer

right, that is what I meant, but I forgot to add the ^2

myininaya · Answer

$$\frac{d}{dx}(-3y)= -3 \frac{d}{dx}y=-3 \frac{dy}{dx}=-3y' \ \frac{d}{dx}(2)=0 \ \frac{d}{dx}[-3x \ln(y)]=-3 \frac{d}{dx}x \ln(y) \text{ here we need product rule }$$

anonymous · Answer

Just to clarify, I'm not having a problem getting there, we understand implicit differentiation. It's simplifying the answer we already have as we cannot have complex fractions or negative exponents

myininaya · Answer

$$\frac{2}{x}-3y'=0-3[\ln(y)+x \frac{y'}{y}]$$

myininaya · Answer

you need to get your y' 's on one side

anonymous · Answer

correct, we already took the derivative and put them on one side. that is what the dy/dx is. the y that is still in the equation is from the derivative of lny

myininaya · Answer

$$\frac{2}{x}+3\ln(y)=-3x \frac{y'}{y}+3y'$$

myininaya · Answer

$$\frac{2}{x}+3 \ln(y)=y'(\frac{-3x}{y}+3)$$
here I would write both the left hand side as one term by combing the fractions
and the thing inside the ( ) on the right hand side as one fraction

myininaya · Answer

then divide both sides by whatever y' is being multiplied by

myininaya · Answer

then get rid of the complex fraction by multiplying bottom and top by the lcm's of their bottoms if you know what I mean

myininaya · Answer

$$\frac{2+3x \ln(y)}{x}=y'(\frac{-3x+3y}{y}) $$

myininaya · Answer

or you could just multiply both sides by the thing that is next to the y' 's reciprocal