Given: $$ \quad y=\frac{1}{4} \sec 4 x $$ Differentiate with the aid of a function of the function rule

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Let $$ y = \frac{1}{4} \sec(4x) $$.

1. Identify the outer function and the inner function:
   - Outer function: $$ f(u) = \frac{1}{4} \sec u $$
   - Inner function: $$ u = 4x $$

2. Differentiate the outer function with respect to $$ u $$:
   $$
   f'(u) = \frac{1}{4} \cdot \sec u \tan u
   $$

3. Differentiate the inner function with respect to $$ x $$:
   $$
   \frac{du}{dx} = 4
   $$

4. Apply the chain rule:
   $$
   \frac{dy}{dx} = f'(u) \cdot \frac{du}{dx} = \left( \frac{1}{4} \sec(4x) \tan(4x) \right) \cdot 4
   $$

5. Simplify the expression:
   $$
   \frac{dy}{dx} = \sec(4x) \tan(4x)
   $$
The derivative of $$ y = \frac{1}{4} \sec(4x) $$ with respect to $$ x $$ is $$ \sec(4x) \tan(4x) $$.

Given:
Differentiate with the aid of a function of the function rule

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Given: Differentiate with the aid of a function of the function rule