Question

In Exercises \(29-32,\) find \(y^{\prime \prime}\) $$y=\cot (3 x-1)$$

Short answer

The second derivative of the function \(y = \cot(3x - 1)\) is \(y^{\prime \prime} = 18 \csc(3x - 1)\cot(3x - 1)\).

Step-by-step solution

Find the first derivative

First, take the first derivative of the function using the chain rule. The derivative of \(y = \cot(x)\) is \(-\csc^2(x)\). Hence the first derivative of \(y = \cot(3x - 1)\) is \(-\csc^2(3x - 1)\) times the derivative of the inner function, which results in \(-3 \csc^2(3x - 1)\).

Calculate the second derivative

The second derivative can be found by taking the derivative of the first derivative. This has to be done carefully as it involves multiple components of trigonometric differentiation and the chain rule. The derivative of \(\csc(x)\) is \(-\csc(x)\cot(x)\), thus the derivative of \(-3 \csc^2(3x - 1)\) is 6 \(\csc(3x - 1) \cot(3x - 1)\) times the derivative of the inner function, which leads to \(y^{\prime \prime} = 18 \csc(3x - 1)\cot(3x - 1)\) as the second derivative.