Question

Consider the curve given by x⁵y⁵ = 250xy. Find the equation of the tangent line to the curve at the point (5, 5).

A) y=x
B) y=2x
C) y= 1/2x
D) y= 1/x

Answer 1

To find the equation of the tangent line to the curve at the point (5, 5), we need to calculate the derivative of the function and then substitute the coordinates of the point into the derivative.

Step-by-step explanation:

To find the equation of the tangent line to the curve at the point (5, 5), we need to calculate the derivative of the function and then substitute the coordinates of the point into the derivative. The given curve is x⁵y⁵ = 250xy. Taking the derivative of both sides with respect to x using the product and chain rule, we get 5x⁴y⁵ + 5x⁵y⁴(dy/dx) = 250y + 250x(dy/dx). Plugging in the coordinates (5, 5), we have 5(5⁴)(5⁵) + 5(5⁵)(5⁴)(dy/dx) = 250(5) + 250(5)(dy/dx). Simplifying this equation, we can solve for dy/dx, which gives us the slope of the tangent line at the given point. The equation of the tangent line can then be written in the form y = mx + b, where m is the slope we found and b is the y-coordinate of the given point.