Question

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the indicated point. $$ x \cos y=1, \quad\left(2, \frac{\pi}{3}\right) $$

Short answer

The derivative of the function at the point \((2, \frac{\pi}{3})\) is \(\sqrt{3}\).

Step-by-step solution

Implicit differentiation

Start by taking the derivative of each side of the equation \(x \cos y = 1\). This results in the equation \((1) \cos y - x \sin y (y') = 0\), where \(y'\) is the derivative of \(y\) with respect to \(x\).

Solving for \(y'\)

Rearrange the equation to solve for \(y'\). The rearranged equation is \(y' = \frac{\cos y}{\sin y}\).

Evaluation at point

Substitute the given point \((2, \frac{\pi}{3})\) into the equation for \(y'\). This results in \(y'(2, \frac{\pi}{3}) = \frac{\cos \frac{\pi}{3}}{\sin \frac{\pi}{3}}\).

Simplify

Simplify the right hand side to obtain the value of the derivative at the point. The simplified value is \(y'(2, \frac{\pi}{3}) = \sqrt{3}\).