Differentiate P(t) = cos⁴(t) + cos(t⁴)

Differentiate each part separately. For cos⁴t, use the chain rule: 4cos³t × (-sin t) = -4 sin t cos³t.

For cos(t⁴), the outer function is cos and the inner is t⁴: -sin(t⁴) × 4t³ = -4t³ sin(t⁴).

Add them: dP/dt = -4(sin t cos³t + t³ sin t⁴). Both terms are negative, so the bracket has a plus sign.