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

To find the derivative of y = 2sin^3 (2x^4 + 1), we can use the chain rule. The chain rule states that if we have a function within another function, we need to differentiate both functions and multiply them together.

Step 1: Differentiate the outer function. The outer function in this case is y = 2sin^3 (2x^4 + 1). To differentiate sin^3 (2x^4 + 1), we can use the power rule. The power rule states that if we have a function raised to a power, we can bring down the power and multiply it by the derivative of the function.

So, differentiating sin^3 (2x^4 + 1) gives us: dy/dx = 3sin^2 (2x^4 + 1) * d/dx (2x^4 + 1)

Step 2: Differentiate the inner function. To differentiate 2x^4 + 1, we can use the power rule again. The derivative of x^n is n*x^(n-1).

So, differentiating 2x^4 + 1 gives us: d/dx (2x^4 + 1) = 8x^3

Step 3: Multiply the derivatives together. Now, we can substitute the derivative of the inner function into the derivative of the outer function.

dy/dx = 3sin^2 (2x^4 + 1) * 8x^3

Simplifying further, we have: dy/dx = 24x^3 * sin^2 (2x^4 + 1)

Therefore, the derivative of y = 2sin^3 (2x^4 + 1) is dy/dx = 24x^3 * sin^2 (2x^4 + 1).