Calculate the derivative of the following functions.
y = cos⁴ θ + sin⁴ θ

Verified step by step guidance

1. Step 1: Recognize that the function y = \cos^4 \theta + \sin^4 \theta is a sum of two terms, each raised to the fourth power. We will need to use the chain rule to differentiate each term separately.
2. Step 2: Apply the chain rule to the first term \cos^4 \theta. The chain rule states that if you have a function u(\theta) raised to a power n, the derivative is n * u(\theta)^{n-1} * u'(\theta). Here, u(\theta) = \cos \theta and n = 4.
3. Step 3: Differentiate \cos \theta with respect to \theta to find u'(\theta). The derivative of \cos \theta is -\sin \theta.
4. Step 4: Combine the results from Steps 2 and 3 to find the derivative of \cos^4 \theta. It will be 4 * \cos^3 \theta * (-\sin \theta).
5. Step 5: Repeat Steps 2-4 for the second term \sin^4 \theta. The derivative of \sin \theta is \cos \theta, so the derivative of \sin^4 \theta is 4 * \sin^3 \theta * \cos \theta. Finally, add the derivatives of both terms to get the derivative of the entire function.