How do you differentiate #arc cot(-4sec(1/(3x^2)) )# using the chain rule?

To differentiate the given function ( \text{arc cot}(-4\sec(\frac{1}{3x^2})) ) using the chain rule, follow these steps:

1. Recognize that the function can be simplified by using the identity ( \sec(\theta) = \frac{1}{\cos(\theta)} ).

2. Rewrite the function as ( \text{arc cot}(-4\cos(\frac{1}{3x^2})^{-1}) ).

3. Apply the chain rule:

   [ \frac{d}{dx}[\text{arc cot}(-4\cos(\frac{1}{3x^2})^{-1})] = -\frac{1}{1 + (-4\cos(\frac{1}{3x^2})^{-1})^2} \times \frac{d}{dx}[-4\cos(\frac{1}{3x^2})^{-1}] ]

4. Find the derivative of the inner function ( -4\cos(\frac{1}{3x^2})^{-1} ):

   [ \frac{d}{dx}[-4\cos(\frac{1}{3x^2})^{-1}] = 4 \sin(\frac{1}{3x^2}) \times \frac{2}{3x^3} ]

5. Substitute the derivative back into the chain rule expression:

   [ \frac{d}{dx}[\text{arc cot}(-4\cos(\frac{1}{3x^2})^{-1})] = -\frac{1}{1 + (-4\cos(\frac{1}{3x^2})^{-1})^2} \times 4 \sin(\frac{1}{3x^2}) \times \frac{2}{3x^3} ]

6. Simplify the expression if necessary.