Question

If \( y = \cos^3(\sec^2 2t) \), find \( \frac{dy}{dt} \).

Hint:

When differentiating trigonometric functions, apply the chain rule carefully.

Answer 1

Step 1: Apply the chain rule for differentiation. The initial expression is: \[ \frac{dy}{dt} = 3\cos^2(\sec^2 2t) \cdot \left[-\sin(\sec^2 2t)\right] \cdot \frac{d}{dt}(\sec^2 2t). \]
Step 2: Compute and substitute the derivative of \( \sec^2 2t \). This derivative is \( \frac{d}{dt}(\sec^2 2t) = 2\sec^2 2t \tan 2t \cdot 2 \). Substituting this back yields: \[ \frac{dy}{dt} = -12 \cos^2(\sec^2 2t) \sin(\sec^2 2t) \sec^2 2t \tan 2t. \] Conclusion: 
The calculated derivative is \( -12 \cos^2(\sec^2 2t) \sin(\sec^2 2t) \sec^2 2t \tan 2t \).