Final answer:
The slope of the curve 7y³+3x⁴=8y+2x at the point (1, 1) is found by differentiating the equation implicitly and evaluating at the given point, resulting in a slope of -10/13.
Step-by-step explanation:
To find the slope of the curve 7y³+3x⁴=8y+2x at the point (1, 1), we need to differentiate this equation implicitly with respect to x to find dy/dx, which represents the slope at any point along the curve. First, let's differentiate each term: the derivative of 7y³ with respect to x is 21y² (dy/dx), the derivative of 3x⁴ is 12x³, the derivative of 8y is 8 (dy/dx), and the derivative of 2x is 2. Arranging the terms gives us:
21y² (dy/dx) - 8 (dy/dx) = 2 - 12x³,
Now simplifying for dy/dx and substituting the given point (1, 1):
dy/dx = (2 - 12x³) / (21y² - 8).
dy/dx = (2 - 12(1)³) / (21(1)² - 8).
dy/dx = (2 - 12) / (21 - 8).
dy/dx = (-10) / (13).
So the slope of the curve at (1, 1) is -10/13.