- #1

Ascendant0

- 57

- 11

- Homework Statement
- Find dy/dx of:

- Relevant Equations
- x^3 = (2x-y)/(x+3y)

I know the *easy* way to do this is shift the (x+3y) to the left side, then differentiate. However, I feel I need to work on my fractions (simplifying them to less complex fractions), plus I wanted to do the more elaborate one to make sure I could do it right, so I instead applied the quotient rule, so for dy/dx:

3x^2 = [(x+3y)(2-dy/dx) - (2x-y)(1+3dy/dx)]/(x+3y)^2

3x^2(x+3y)^2 = 2x-xdy/dx+6y-3ydy/dx - 2x-6xdy/dx+y+3ydy/dx

3x^2(x^2+6xy+9y^2) = -7xdy/dx+7y

3x^4+18yx^3+27(xy)^2 = -7xdy/dx+7y

[3x^4+18yx^3+27(xy)^2-7y]/(-7x) = dy/dx

But this answer isn't right at all, as if I plug the same values for x and y in this answer, vs the one from the solutions manual, I get different values (their answer is (2-4x^3-9yx^2)/(3x^3+1). If I use a simple x = y = 1 in my above answer and this answer, the values are completely different (~-5.857 for mine, their answer is -2.75).

And I worked this problem by shifting "x+3y" to the left first and then finding the derivative, and I got the same answer as the solution manual, so I know I'm messing this up somewhere when I use the quotient rule. But I've literally done it five times now, and I've gotten the same exact answer all five times. It's driving me crazy. Can someone PLEASE tell me where I'm messing up, as I need to know, because it's an error I'm consistently making and need to fix it for future problems.

3x^2 = [(x+3y)(2-dy/dx) - (2x-y)(1+3dy/dx)]/(x+3y)^2

3x^2(x+3y)^2 = 2x-xdy/dx+6y-3ydy/dx - 2x-6xdy/dx+y+3ydy/dx

3x^2(x^2+6xy+9y^2) = -7xdy/dx+7y

3x^4+18yx^3+27(xy)^2 = -7xdy/dx+7y

[3x^4+18yx^3+27(xy)^2-7y]/(-7x) = dy/dx

But this answer isn't right at all, as if I plug the same values for x and y in this answer, vs the one from the solutions manual, I get different values (their answer is (2-4x^3-9yx^2)/(3x^3+1). If I use a simple x = y = 1 in my above answer and this answer, the values are completely different (~-5.857 for mine, their answer is -2.75).

And I worked this problem by shifting "x+3y" to the left first and then finding the derivative, and I got the same answer as the solution manual, so I know I'm messing this up somewhere when I use the quotient rule. But I've literally done it five times now, and I've gotten the same exact answer all five times. It's driving me crazy. Can someone PLEASE tell me where I'm messing up, as I need to know, because it's an error I'm consistently making and need to fix it for future problems.