1. Stated Problem: Differentiate the function $$H(x) = 3 \sec x (1 - \tan x)$$. 2. Use the product rule for differentiation: If $$H(x) = f(x)g(x)$$, then $$H'(x) = f'(x)g(x) + f(x)g'(x)$$. 3. Identify $$f(x) = 3 \sec x$$ and $$g(x) = 1 - \tan x$$. 4. Differentiate each function: - $$f'(x) = 3 \sec x \tan x$$ (since derivative of $$\sec x$$ is $$\sec x \tan x$$) - $$g'(x) = -\sec^2 x$$ (since derivative of $$\tan x$$ is $$\sec^2 x$$) 5. Apply the product rule: $$H'(x) = 3 \sec x \tan x (1 - \tan x) + 3 \sec x (-\sec^2 x)$$ 6. Simplify the expression: $$H'(x) = 3 \sec x \tan x - 3 \sec x \tan^2 x - 3 \sec^3 x$$ 7. Final answer: $$\boxed{H'(x) = 3 \sec x \tan x - 3 \sec x \tan^2 x - 3 \sec^3 x}$$