Find dy/dx by implicit differentiation

1. x^3 -xy+ y^2= 7

The given answer is (y-3x^2)/ (2y-x), but I'm not getting it.

2. How would I do this one? If I couldn't do the above one, which looks a heck of a lot simpler, I doubt I'll get through this one.

x^ (6/7) + y^ (8/5) = 16

x^3 -xy+ y^2= 7

3x^2-xdy/dx-y+2ydy/dx=0

dy/dx(2y-x)=y-3x^2

Why is it you cannot get the answer?

x^ (6/7) + y^ (8/5) = 16
6/7 x^(-2/7) + 8/5 y^(3/5)y'=0

y'= (-6/7 x^(-2/7))/(8/5 y^)3/5))

do you agree?

you may need to adjust for the fact that 6/7 - 1 = -1/7, not -2/7

To find dy/dx using implicit differentiation, follow these steps:

1. Differentiate both sides of the equation with respect to x.
2. Treat y as a function of x and apply the chain rule whenever necessary.
3. After differentiating, collect all terms involving dy/dx on one side.
4. Solve for dy/dx.

Let's apply these steps to the given equations:

1. x^3 - xy + y^2 = 7

Step 1: Differentiate both sides with respect to x.
d/dx(x^3 - xy + y^2) = d/dx(7)
3x^2 - x(dy/dx) + 2yy' = 0

Step 2: Apply the chain rule.
For the term -xy, differentiate each variable separately, treating the other as a constant.
-xy = -x(dy/dx) - y

For the term y^2, differentiate using the chain rule.
d/dx(y^2) = 2y(dy/dx)

So, the equation becomes:
3x^2 - x(dy/dx) + 2yy' = 2y(dy/dx) - x(dy/dx) - y

Step 3: Collect the terms.
Rearrange the equation to isolate dy/dx terms on one side:
- x(dy/dx) - x(dy/dx) + 2yy' = 2y(dy/dx) - 3x^2 - y

Step 4: Solve for dy/dx.
Combine like terms:
- 2x(dy/dx) + 2yy' = 2y(dy/dx) - 3x^2 - y

Rearrange terms involving dy/dx on one side:
2yy' - 2y(dy/dx) = -3x^2 + y
-2y(dy/dx) - 2yy' = -3x^2 + y
-(2y + 2yy') = -3x^2 + y

Finally, isolate dy/dx:
dy/dx = (y - 3x^2) / (2y - x)

So, the answer to the first equation is indeed (y - 3x^2) / (2y - x).

Let's move on to the second equation:

2. x^(6/7) + y^(8/5) = 16

You can follow the same steps to find dy/dx:

Step 1: Differentiate both sides with respect to x.
d/dx(x^(6/7) + y^(8/5)) = d/dx(16)
(6/7)x^(-1/7) + (8/5)y^(-7/5)(dy/dx) = 0

Step 2: Apply the chain rule.
For the term x^(6/7), differentiate using the power rule for differentiation.
d/dx(x^(6/7)) = (6/7)x^(-1/7)

For the term y^(8/5), differentiate using the chain rule.
d/dx(y^(8/5)) = (8/5) y^(-2/5)(dy/dx)

So, the equation becomes:
(6/7)x^(-1/7) + (8/5)y^(-2/5)(dy/dx) = 0

Step 3: Collect the terms.
Rearrange the equation:
(6/7)x^(-1/7) + (8/5)y^(-2/5)(dy/dx) = 0

Step 4: Solve for dy/dx.
Move the term involving dy/dx to the right side:
(8/5)y^(-2/5)(dy/dx) = - (6/7)x^(-1/7)

Isolate dy/dx:
dy/dx = - (6/7)x^(-1/7) / (8/5)y^(-2/5)

Simplify the expression:
dy/dx = - (30/56) (x^(-1/7)) / (y^(-2/5))

Note: It is common to rationalize the expression by multiplying the numerator and denominator by the conjugate of the denominator. However, in this case, multiplying by the conjugate does not simplify the expression further.

So, the answer to the second equation is dy/dx = - (30/56) (x^(-1/7)) / (y^(-2/5))