Calculus slope question

Question

xy^5/2+x^3/2y=80;(16,1) what is the slope of the curve

Answer 1

The slope is nothing but the gradient (first derivative) of the curve.

Use implicit differentiation and the product rule.

(5/2)y^3/2xdy/dx + y^5/2 + x^3/2dy/dx + (3/2)x^1/2y = 0

grouping like terms and simplifying gives

dy/dx = (-3x^1/2y - 2y^5/2)/(5y^3/2x + 2x^3/2) plugging in the values of x, and y i.e. {x=16,y=1} gives the slope you want.