Final answer:
To find dy/dx using implicit differentiation for the equation (2x³ - 4y²)³ = -xy, apply the chain rule and product rule, differentiate with respect to x, collect terms with dy/dx, and solve for dy/dx to express it in terms of x and y.
Step-by-step explanation:
The question involves applying implicit differentiation to find the derivative dy/dx for the equation (2x³ – 4y²)³ = -xy. To solve this problem, we first apply the chain rule to differentiate both sides of the equation with respect to x. When differentiating the left side, treat y as a function of x (y(x)), which requires us to use the chain rule and product rule for differentiation.
We begin by differentiating the outer function, which is the cube of the inner function. Remember to keep the inner function (2x³ - 4y²) intact. Next, differentiate the inner function with respect to x. For the x terms, direct differentiation is used, and for terms involving y, we multiply by dy/dx because y is a function of x. On the right side of the equation, apply the product rule to differentiate -xy.
After differentiating, you will have an equation that includes terms with dy/dx. Collect all terms with dy/dx on one side and move the rest to the other side. Factor out dy/dx and solve for dy/dx by dividing both sides by the expression that is multiplied by dy/dx. This will give you the value of dy/dx in terms of x and y.