find dy/dx
-x^2y+2xy=3y^2-7
To find dy/dx, we need to use implicit differentiation. This involves differentiating both sides of the equation with respect to x and then solving for dy/dx.
Let's differentiate each term separately:
Differentiating -x^2y:
The derivative of -x^2y with respect to x can be found using the product rule. The product rule states that if we have a function f(x) = u(x) * v(x), then the derivative of f(x) with respect to x is given by f'(x) = u'(x) * v(x) + u(x) * v'(x).
In this case, u(x) = -x^2 and v(x) = y. So, applying the product rule, we have:
d/dx(-x^2y) = -2xy - x^2 * dy/dx.
Differentiating 2xy:
Since x and y are both variables, we can treat them as separate functions and use the product rule again. Here, u(x) = 2x and v(x) = y. Applying the product rule, we get:
d/dx(2xy) = 2y + 2x * dy/dx.
Differentiating 3y^2:
The derivative of 3y^2 with respect to x is 0 because y^2 is a constant with respect to x.
Differentiating -7:
The derivative of a constant term (-7 in this case) with respect to x is 0.
Now, let's put it all together and solve for dy/dx:
-2xy - x^2 * dy/dx + 2y + 2x * dy/dx = 0 (combining the derivatives)
Rearranging the terms:
- x^2 * dy/dx + 2x * dy/dx = -2xy - 2y
Factoring out dy/dx:
dy/dx * (- x^2 + 2x) = -2xy - 2y
Dividing both sides by (- x^2 + 2x):
dy/dx = (-2xy - 2y) / (- x^2 + 2x)
And that's the expression for dy/dx.