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Section Assignment #7 - Done In Class on Thursday 3/6

  1. Use implicit differentiation to find \(\frac{dy}{dx}\) for the implicit equation \[ (x-y)^2=x+y-1. \]
    1. Find the derivative of the left side of the equation, with respect to x.
      2. \[ \frac{d}{dx}(x-y)^2=2(x-y)\frac{d}{dx}(x-y)=2(x-y)(1-\frac{dy}{dx}). \]
    2. Find the derivative of the right side of the equation, with respect to x.
      3. \[ \frac{d}{dx} x+y-1=1+\frac{dy}{dx} \]
    3. Set the derivatives equal to each other and solve for \(\frac{dy}{dx}\):
      4. \( \begin{align*} 2(x-y)(1-\frac{dy}{dx})&= 1+\frac{dy}{dx}\\ \implies & 2(x-y)-2(x-y)\frac{dy}{dx}=1+\frac{dy}{dx}\\ \implies & 2(x-y)-1=\frac{dy}{dx}(1+2(x-y))\\ \implies & \frac{dy}{dx}=\frac{2x-2y-1}{1+2x-2y}. \end{align*} \)