Find the derivative of y = 2sin^3 (2x^4 + 1)*72x^3 sin^2 (2x^4 + 1) cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1) sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1) sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1) cos⁡(2x^4 + 1)

To find the derivative of the given function, we will use the chain rule and product rule. Let's break it down step by step:

Step 1: Identify the function and its components. The given function is y = 2sin^3 (2x^4 + 1)*72x^3 sin^2 (2x^4 + 1) cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1) sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1) sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1) cos⁡(2x^4 + 1)

Step 2: Apply the chain rule. The chain rule states that if we have a composite function, we need to differentiate the outer function with respect to the inner function, and then multiply it by the derivative of the inner function.

Step 3: Differentiate the outer function. The outer function is 2sin^3 (2x^4 + 1)*72x^3 sin^2 (2x^4 + 1) cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1) sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1) sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1) cos⁡(2x^4 + 1)

Step 4: Differentiate the inner function. The inner function is (2x^4 + 1).

Step 5: Apply the product rule. The product rule states that if we have a product of two functions, we need to differentiate each function separately and then add them together.

Step 6: Differentiate each function separately. The first function is 2sin^3 (2x^4 + 1)*72x^3 sin^2 (2x^4 + 1) cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1) sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1) sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1) cos⁡(2x^4 + 1) The second function is (2x^4 + 1).

Step 7: Differentiate each function separately. Differentiating the first function requires the use of the chain rule and the product rule.

Step 8: Apply the chain rule to the first function. Differentiate the outer function (2sin^3 (2x^4 + 1)) with respect to the inner function (2x^4 + 1).

Step 9: Differentiate the outer function. The outer function is 2sin^3 (2x^4 + 1).

Step 10: Differentiate the inner function. The inner function is (2x^4 + 1).

Step 11: Apply the product rule to the first function. Differentiate each term separately and then add them together.

Step 12: Differentiate each term separately. The first term is 2sin^3 (2x^4 + 1). The second term is 72x^3 sin^2 (2x^4 + 1). The third term is cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1). The fourth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The fifth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The sixth term is cos⁡(2x^4 + 1).

Step 13: Differentiate each term separately. Differentiating each term requires the use of the chain rule and the product rule.

Step 14: Apply the chain rule to the first term. Differentiate the outer function (2sin^3 (2x^4 + 1)) with respect to the inner function (2x^4 + 1).

Step 15: Differentiate the outer function. The outer function is 2sin^3 (2x^4 + 1).

Step 16: Differentiate the inner function. The inner function is (2x^4 + 1).

Step 17: Apply the product rule to the first term. Differentiate each term separately and then add them together.

Step 18: Differentiate each term separately. The first term is 2sin^3 (2x^4 + 1). The second term is 72x^3 sin^2 (2x^4 + 1). The third term is cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1). The fourth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The fifth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The sixth term is cos⁡(2x^4 + 1).

Step 19: Differentiate each term separately. Differentiating each term requires the use of the chain rule and the product rule.

Step 20: Apply the chain rule to the first term. Differentiate the outer function (2sin^3 (2x^4 + 1)) with respect to the inner function (2x^4 + 1).

Step 21: Differentiate the outer function. The outer function is 2sin^3 (2x^4 + 1).

Step 22: Differentiate the inner function. The inner function is (2x^4 + 1).

Step 23: Apply the product rule to the first term. Differentiate each term separately and then add them together.

Step 24: Differentiate each term separately. The first term is 2sin^3 (2x^4 + 1). The second term is 72x^3 sin^2 (2x^4 + 1). The third term is cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1). The fourth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The fifth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The sixth term is cos⁡(2x^4 + 1).

Step 25: Differentiate each term separately. Differentiating each term requires the use of the chain rule and the product rule.

Step 26: Apply the chain rule to the first term. Differentiate the outer function (2sin^3 (2x^4 + 1)) with respect to the inner function (2x^4 + 1).

Step 27: Differentiate the outer function. The outer function is 2sin^3 (2x^4 + 1).

Step 28: Differentiate the inner function. The inner function is (2x^4 + 1).

Step 29: Apply the product rule to the first term. Differentiate each term separately and then add them together.

Step 30: Differentiate each term separately. The first term is 2sin^3 (2x^4 + 1). The second term is 72x^3 sin^2 (2x^4 + 1). The third term is cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1). The fourth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The fifth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The sixth term is cos⁡(2x^4 + 1).

Step 31: Differentiate each term separately. Differentiating each term requires the use of the chain rule and the product rule.

Step 32: Apply the chain rule to the first term. Differentiate the outer function (2sin^3 (2x^4 + 1)) with respect to the inner function (2x^4 + 1).

Step 33: Differentiate the outer function. The outer function is 2sin^3 (2x^4 + 1).

Step 34: Differentiate the inner function. The inner function is (2x^4 + 1).

Step 35: Apply the product rule to the first term. Differentiate each term separately and then add them together.

Step 36: Differentiate each term separately. The first term is 2sin^3 (2x^4 + 1). The second term is 72x^3 sin^2 (2x^4 + 1). The third term is cos⁡(2x^4 + 1)72x^3 sin^2 (2x^4 + 1). The fourth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The fifth term is sin⁡(2x^4 + 1)72x^3 cos^2 (2x^4 + 1). The sixth term is cos⁡(2x^4 + 1).

Step 37: Differentiate each term separately. Differentiating each term requires the use of the chain rule and the product rule.

Step 38: Apply the chain rule to the first term. Differentiate the outer function (2sin^3 (2x^4 + 1)) with respect to the inner function (2x^4 + 1).

Step 39: Differentiate the outer function. The outer function is 2sin^3 (2x^4 + 1).

Step 40: Differentiate the inner function. The inner function is (2x^4 + 1).

Step 41: Apply the product rule to the first term. Differentiate each term separately and then add them together.

Step 42: Differentiate each term separately. The first term