Question

A curve is defined by the equation (x+y)^2-3xy=9. Find the rate of change of the slope of the curve at the point (0,3)?

Answer 1

by rate of change of slope i assume you mean the 2nd derivative

differentiate and find 1st derivative
\[2(x+y)(1+y') - 3y -3xy' = 0\]
\[y' = \frac{y-2x}{2y-x}\]
Next take 2nd derivative using quotient rule
\[y'' = \frac{(y'-2)(2y-x) - (2y'-1)(y-2x)}{(2y-x)^2}\]
Finally plug in given point (0,3) to obtain value for y' and then using y' , find y''
\[y' = \frac{1}{2}\]
\[y'' = -\frac{1}{4}\]