EXAMPLE 2 (a) Find y' if x3 + y3 = 18xy. (b) Find the tangent to the folium of Descartes x3 + y3 = 18xy at the point (9, 9). (c) At what point in the first quadrant is the tangent line horizontal?

(a) Via implicit differentiation, we get

Solve for :

(b) Find the slope of the tangent line at (9, 9) by plugging in x = y = 9 into the equation above:

Use the point-slope formula to find the equation of the line:

(c) The tangent line is horizontal when its slope is 0, so solve :

Now substitute y in the equation for the folium to solve for x :

x = 0 corresponds to y = 0 (plug x = 0 into the folium equation to see why), i.e. the origin. If you don't consider the origin to belong to the first quadrant, then we only keep

let f (x ) is a function and c be a positive real number,   the graph of the function g (x ) = f (cx ) is represented as follows.

when, c > 1,   the function compresses it in the x - direction.

when, 0 < c < 1, the function stretches it.

the function is g ( x ) = (9x )2.

the heighest power of x in g ( x ) = (9x )2 is 2, so the parent function is square function.

the equation of parent function is f ( x ) = x 2 .

the function is g ( x ) = (9x )2 and the parent function is f ( x ) = x 2 .

so, the functions f and g have the following relationship.

g ( x ) = f ( 9x ).

compare the function g ( x ) = f ( 9x ) with g (x ) = f (cx ).

c = 9 ( > 1 ).

so, the function g (x ) = (9x )2 compresses it in the x - direction.

step-by-step explanation:

the answer to this would be 5x^3-6x^2-24 over 3x^2

hope this if needed hmu