Calculate the derivative of the following functions.
y = (2x⁶ - 3x³ + 3)²⁵

Verified step by step guidance

1. Step 1: Recognize that the function y = (2x^6 - 3x^3 + 3)^25 is a composite function, which means we will need to use the chain rule to find its derivative.
2. Step 2: Identify the outer function and the inner function. Here, the outer function is u^25, where u = 2x^6 - 3x^3 + 3, and the inner function is u = 2x^6 - 3x^3 + 3.
3. Step 3: Differentiate the outer function with respect to the inner function u. The derivative of u^25 with respect to u is 25u^24.
4. Step 4: Differentiate the inner function u with respect to x. The derivative of 2x^6 is 12x^5, the derivative of -3x^3 is -9x^2, and the derivative of the constant 3 is 0.
5. Step 5: Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us the derivative of y with respect to x as dy/dx = 25(2x^6 - 3x^3 + 3)^24 * (12x^5 - 9x^2).