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 equation 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.
The derivative of $\sec^2 2t$ is:
\[\frac{d}{dt}(\sec^2 2t) = 2\sec^2 2t \tan 2t \cdot 2.\]Substituting this back into the main equation yields:
\[\frac{dy}{dt} = -12 \cos^2(\sec^2 2t) \sin(\sec^2 2t) \sec^2 2t \tan 2t.\]
Conclusion: The final derivative is \( -12 \cos^2(\sec^2 2t) \sin(\sec^2 2t) \sec^2 2t \tan 2t \).