009A Sample Final 1, Problem 7 Detailed Solution

A curve is defined implicitly 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  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{3}+y^{3}=6xy}   at the point  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (3,3)} .

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Background Information:
1. What is the result of implicit differentiation of  $xy?$
It would be  $y+x{\frac {dy}{dx}}$  by the Product Rule.
2. What two pieces of information do you need to write the equation of a line?
You need the slope of the line and a point on the line.
3. What is the slope of the tangent line of a curve?
The slope is  $m={\frac {dy}{dx}}.$

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 to get
${\frac {dy}{dx}}={\frac {3x^{2}-6y}{6x-3y^{2}}}.$

(b)

Step 1:
First, we find the slope of the tangent line at the point  $(3,3).$
We plug  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (3,3)}   into the formula for  ${\frac {dy}{dx}}$  we found in part (a).
So, we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{m} & = & \displaystyle{\frac{3(3)^2-6(3)}{6(3)-3(3)^2}}\\ &&\\ & = & \displaystyle{-\frac{9}{9}}\\ &&\\ & = & \displaystyle{-1.} \end{array}}

Step 2:
Now, we have the slope of the tangent line at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3,3)}   and a point.
Thus, we can write the equation of the line.
So, the equation of the tangent line at  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3,3)}   is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\,=\,-1(x-3)+3.}

Final Answer:
(a)    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}=\frac{3x^2-6y}{6x-3y^2}}
(b)    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=-1(x-3)+3}

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