Final answer:
To differentiate the function (a) f(x) = (2x^2 + 1)^2023, use the chain rule, and for (b) y = sqrt(3x^2 + 8x + 9), use the power rule.
Step-by-step explanation:
To differentiate the function f(x) = (2x^2 + 1)^2023, we can use the chain rule. The chain rule states that if we have a composition of functions, the derivative of the composition can be found by multiplying the derivative of the outer function with the derivative of the inner function. In this case, the outer function is raising to the power of 2023, and the inner function is 2x^2 + 1. Applying the chain rule, we get:
f'(x) = 2023(2x^2 + 1)^2022(4x)
Now, let's differentiate the function y = sqrt(3x^2 + 8x + 9). To do this, we can use the power rule, since the square root function can be written as raising to the power of 1/2. The power rule states that if we have a function raised to a power, the derivative can be found by bringing down the power and multiplying it with the derivative of the function. In this case, the function inside the square root is 3x^2 + 8x + 9. Applying the power rule, we get:
y' = (1/2)(3x^2 + 8x + 9)^(-1/2)(6x + 8)