use implicit differentiation to find dy/dx if x^3+y^3=6xy

3x^2 + 3y^2 dy/dx = 6x dy/dx + 6y

dy/dx(3y^2 - 6x) = 6y - 3x^2

dy/dx = (6y - 3x^2)/(3y^2 - 6x)
= (2y - x^2)/(y^2 - 2x)

To find dy/dx using implicit differentiation, we will differentiate both sides of the equation with respect to x while treating y as a function of x.

Given: x^3 + y^3 = 6xy

Differentiating both sides of the equation with respect to x:

d/dx [x^3 + y^3] = d/dx [6xy]

Using the power rule, the derivative of x^n with respect to x is nx^(n-1), we get:

3x^2 + d/dx [y^3] = 6(d/dx [xy])

We need to find d/dx [y^3].

Using the chain rule, d/dx [f(g(x))] = f'(g(x)) * g'(x), where f(u) = u^3 and g(x) = y(x).

Applying the chain rule, we get:

d/dx [y^3] = 3y^2 * d/dx[y]

Now we substitute this back into the equation:

3x^2 + 3y^2 * d/dx[y] = 6(d/dx[xy])

Since x and y are both variables, we can rearrange the equation to isolate dy/dx:

3y^2 * d/dx[y] = 6(d/dx[xy]) - 3x^2

Now we need to find d/dx[xy]. We use the product rule, which states that d/dx[u * v] = u * d/dx[v] + v * d/dx[u].

Using the product rule, we get:

d/dx[xy] = x * d/dx[y] + y * d/dx[x]

Since d/dx[x] = 1 (the derivative of x with respect to x is 1), and d/dx[y] is what we're looking for, we can rewrite this as:

d/dx[xy] = x * dy/dx + y

Substituting this back into the equation:

3y^2 * d/dx[y] = 6(x * dy/dx + y) - 3x^2

Now we can solve for dy/dx:

3y^2 * d/dx[y] = 6x * dy/dx + 6y - 3x^2

Isolating dy/dx, we get:

3y^2 * d/dx[y] - 6x * dy/dx = 6y - 3x^2

Now we divide by (3y^2 - 6x) to solve for dy/dx:

dy/dx = (6y - 3x^2) / (3y^2 - 6x)

Finally, we have found the derivative dy/dx using implicit differentiation for the given equation x^3 + y^3 = 6xy.

To find the derivative of y with respect to x, dy/dx, using implicit differentiation for the equation x^3 + y^3 = 6xy, follow these steps:

Step 1: Differentiate both sides of the equation with respect to x.
(Note: Treat y as a function of x and use the chain rule when differentiating y terms.)

Differentiating x^3 with respect to x gives 3x^2.
Differentiating y^3 with respect to x requires the chain rule: d/dx(y^3) = 3y^2(dy/dx).

To differentiate the right side, we use the product rule. Differentiating 6xy will give 6x(dy/dx) + 6y.

So, we have: 3x^2 + 3y^2(dy/dx) = 6x(dy/dx) + 6y.

Step 2: Simplify the equation by grouping terms with dy/dx together on one side and the remaining terms on the other side.

Move the terms containing dy/dx to the left side and the other terms to the right side:

3y^2(dy/dx) - 6x(dy/dx) = 6x - 3x^2 - 6y.

Factor out dy/dx on the left side:

(dy/dx)(3y^2 - 6x) = 6x - 3x^2 - 6y.

Step 3: Solve for dy/dx by dividing both sides by (3y^2 - 6x).

dy/dx = (6x - 3x^2 - 6y) / (3y^2 - 6x).

Therefore, dy/dx = (6x - 3x^2 - 6y) / (3y^2 - 6x) is the derivative of y with respect to x using implicit differentiation for the equation x^3 + y^3 = 6xy.