Question

Find the slope of the graph of the relation y^3 − xy = −6 at the point (7, 2).

Answer 1

Yay, implicit differnentiation

when you take the derivitive of y, you multiply it by dy/dx
example
dy/dx y^2=2y dy/dx

for x, the dy/dx dissapears

ok
so differnetiate and solve for dy/dx

3y² dy/dx-(y+x dy/dx)=0
expand
3y² dy/dx-y-x dy/dx=0
3y² dy/dx-x dy/dx=y
dy/dx (3y²-x)=y
dy/dx=y/(3y²-x)

so at (7,2)
x=7 and y=2
dy/dx=2/(3(2)²-7)
dy/dx=2/(3(4)-7)
dy/dx=2/(12-7)
dy/dx=2/5

answer is 2/5