009A Sample Final 1, Problem 7

A curve is defined implicityly by the equation

$x^{3}+y^{3}=6xy$

a) Using implicit differentiation, compute ${\frac {dy}{dx}}$.

b) Find an equation of the tangent line to the curve $x^{3}+y^{3}=6xy$ at the point $(3,3)$.

Foundations:

Solution:

(a)

Step 1:
Using implicit differentiation on the equation $x^{3}+y^{3}=6xy$, we get
$3x^{2}+3y^{2}{\frac {dy}{dx}}=6y+6x{\frac {dy}{dx}}$.

Step 2:
Now, we move all the ${\frac {dy}{dx}}$ terms to one side of the equation.
So, we have
$3x^{2}-6y={\frac {dy}{dx}}(6x-3y^{2})$.
We solve for ${\frac {dy}{dx}}={\frac {3x^{2}-6y}{6x-3y^{2}}}$.

(b)

Step 1:

Step 2:

Step 3:

Final Answer:
(a)
(b)

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