Question: Find the derivative of the equation 4x^2 + 5xy - 3y^2 = 20 with respect to x using implicit differentiation.

Answer: To find the derivative of the given equation with respect to x using implicit differentiation, we differentiate each term with respect to x and then solve for dy/dx.

Given equation: 4x^2 + 5xy - 3y^2 = 20

Differentiating each term with respect to x: d/dx(4x^2) + d/dx(5xy) - d/dx(3y^2) = d/dx(20)

Applying the power rule and product rule, we have: 8x + 5y + 5x(dy/dx) - 6y(dy/dx) = 0

Now we can solve for dy/dx: Collecting terms with dy/dx on one side, 5x(dy/dx) - 6y(dy/dx) = -8x - 5y

Factor out dy/dx, dy/dx(5x - 6y) = -8x - 5y

Divide by (5x - 6y) to solve for dy/dx, dy/dx = (-8x - 5y) / (5x - 6y)

Therefore, the derivative of the given equation with respect to x using implicit differentiation is dy/dx = (-8x - 5y) / (5x - 6y).