009A Sample Final 2, Problem 4

Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3x^{2}+xy+y^{2}=5}   at the point  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (1,-2)}

Foundations:
The equation of the tangent line to  $f(x)$  at the point  $(a,b)$  is
$y=m(x-a)+b$  where  $m=f'(a).$

Solution:

Step 1:
We use implicit differentiation to find the derivative of the given curve.
Using the product and chain rule, we get
$6x+xy'+y+2yy'=5.$
We rearrange the terms and solve for  $y'.$
Therefore,
$xy'+2yy'=5-6x-y$
and
$y'={\frac {5-6x-y}{x+2y}}.$

Step 2:
Therefore, the slope of the tangent line at the point  $(1,-2)$  is
${\begin{array}{rcl}\displaystyle {m}&=&\displaystyle {\frac {5-6(1)-(-2)}{1-4}}\\&&\\&=&\displaystyle {{\frac {1}{-3}}.}\end{array}}$
Hence, the equation of the tangent line to the curve at the point  $(1,-2)$  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 f(x)=-\frac{1}{3}(x-1)-2.}

Final Answer:
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 f(x)=-\frac{1}{3}(x-1)-2.}

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