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Chapter 2: Problem 16

Find \(d y / d x\) by implicit differentiation. $$ \cot y=x-y $$

Short Answer

Expert verified
Therefore, the derivative \(dy/dx\) is \( \sin^2 y / (\sin^2 y + 1)\).

Step by step solution

01

Differentiate both sides with respect to \(x\)

Differentiating both sides of the equation \(\cot y=x-y\) with respect to \(x\) gives \( -\csc^{2} y (dy/dx) = 1 - dy/dx\). Here, \(\csc y\) is the cosecant of \(y\), equal to \(1/\sin y\), and the derivative of \(\cot y\) with respect to \(y\) is \(-\csc^{2}y\). On the right side, the derivative of \(x\) with respect to \(x\) is 1, and the derivative of \(-y\) with respect to \(x\) is \(-dy/dx\).
02

Solve for \(dy/dx\)

Rearrange the equation to solve for \(dy/dx\). Initially, this means consolidating terms containing \(dy/dx\) on one side of the equation and the constants on the other. This gives us \(- \csc^{2} y (dy/dx) + dy/dx = 1\). Factor the \(dy/dx\) from the left side to get \(dy/dx = 1 / (1 + \csc^{2} y)\) which simplifies to \(dy/dx = 1 / (1 +(1/ \sin^2 y))\).
03

Further Simplify the \(dy/dx\) expression

Combine the fractions on the right side of the equation to simplify the expression, resulting in \(dy/dx = \sin^2 y / (\sin^2 y + 1)\).

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