Final answer:
To find the slope of the curve at the given point (1,1), we need to take the derivative of the equation and evaluate it at the given point. The slope of the curve at (1,1) is -35/61.
Step-by-step explanation:
To find the slope of the curve at a given point, we need to take the derivative of the equation and evaluate it at the given point. The given equation is 8y⁸+5x³=3y+10x. We can rewrite this equation as 8y⁸ - 3y = -5x³ - 10x. Taking the derivative with respect to x, we get 64y⁷(dy/dx) - 3(dy/dx) = -15x² - 10. Plugging in the values of x and y from the given point (1,1), we can solve for the derivative, which represents the slope of the curve at that point.
Plugging in x = 1 and y = 1 into the derivative equation, we get 64(1)⁷(dy/dx) - 3(dy/dx) = -15(1)² - 10. Simplifying this equation, we have 64(dy/dx) - 3(dy/dx) = -25 - 10. Combining like terms, we get 61(dy/dx) = -35. Dividing both sides by 61, we find that dy/dx = -35/61, which is the slope of the curve at the point (1,1).