022 Exam 1 Sample A, Problem 2

2. Use implicit differentiation to find $dy/dx$ at the point $(1,0)$ on the curve defined by $x^{3}-y^{3}-y=x$.

Foundations:
When we use implicit differentiation, we combine the chain rule with the fact that $y$ is a function of $x$, and could really be written as $y(x).$ Because of this, the derivative of $y^{3}$ with respect to $x$ requires the chain rule, so
${\frac {d}{dx}}\left(y^{3}\right)=3y^{2}\cdot {\frac {dy}{dt}}.$

 Solution:

Step 1:
First, we differentiate each term separately with respect to $x$ to find that  $x^{3}-y^{3}-y=x$  differentiates implicitly to
$3x^{2}-3y^{2}\cdot {\frac {dy}{dx}}-{\frac {dy}{dx}}=1$.

Step 2:
Since they don't ask for a general expression of 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 dy/dx} , but rather a particular value at a particular point, we can plug in the values 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 x=1} and 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=0}  to find
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(1)^{2}-3(0)^{2}\cdot\frac{dy}{dx}-\frac{dy}{dx}=1,}
which is equivalent to 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-\frac{dy}{dx}=1} . This solves to 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 dy/dx=2.}

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