Final answer:
To find the equation of the tangent line to the curve at the point (5, 5), we need to find the slope of the curve at that point. The slope of the curve at the point (5, 5) is 1/2. Therefore, the equation of the tangent line is y - 5 = (1/2)(x - 5).
Step-by-step explanation:
To find the equation of the tangent line to the curve at the point (5, 5), we need to find the slope of the curve at that point.
To do this, we can take the derivative of the curve equation with respect to x. Taking the derivative of both sides of the equation, we get:
5x^4 + 5y^4*(dy/dx) = 250*(dy/dx)*y + 250xy'
Plugging in the coordinates of the point (5, 5), we get:
5*(5)^4 + 5*(5)^4*(dy/dx) = 250*(dy/dx)*5 + 250*(5)*(dy/dx)
Simplifying, we have:
625 + 625*(dy/dx) = 1250*(dy/dx) + 1250*(dy/dx)
Dividing both sides by 1250*(dy/dx), we get:
625/1250 = (dy/dx)
Simplifying further, we have:
1/2 = (dy/dx)
Therefore, the slope of the curve at the point (5, 5) is 1/2.
Since the tangent line has the same slope as the curve at that point, the equation of the tangent line is y - 5 = (1/2)(x - 5).