Final answer:
To find the implicit differentiation of x^2-2xy⋅y^2=y, differentiate both sides of the equation with respect to x. Apply the power rule and product rule to differentiate each term, and simplify the result.
Step-by-step explanation:
To find the implicit differentiation of x² - 2xy⋅y² = y, we need to differentiate both sides of the equation with respect to x. Let's start with the left side:
d/dx (x² - 2xy⋅y²)
To differentiate x², we use the power rule, which states that the derivative of x^n is n⋅x^(n-1). So, the derivative of x² is 2x. For the second term, we use the product rule:
d/dx (-2xy⋅y²)
Using the product rule, we differentiate the first term (-2xy) with respect to x, which gives us -2y, and then we multiply it by the derivative of the second term (y²): -2y⋅2y^2 = -4y^3.
So, d/dx (-2xy⋅y²) = -4y^3.
Putting it all together, we have:
d/dx (x² - 2xy⋅y²) = 2x - 4y^3 = d/dx (y).