Final answer:
The derivative of the function f(x) = 3x + (3x + (3x + 1)⁴)⁴ can be found by applying the chain rule multiple times and simplifying the result accordingly.
Step-by-step explanation:
To calculate the derivative of the function f(x) = 3x + (3x + (3x + 1)⁴)⁴, we will apply the chain rule multiple times due to the nested functions present.
First, let's differentiate the outermost layer: (3x + something)⁴. We will treat 'something' as a function of x that we'll call g(x). The derivative is 4(3x + g(x))³ × (3 + g'(x)).
Now, let's differentiate g(x) = 3x + (3x + 1)⁴. This will be 3 + 4(3x + 1)³ × 3.
Combining these expressions and simplifying gives us:
f'(x) = 3 + 4(3x + g(x))³ × (3 + 4(3x + 1)³ × 3)
Finally, apply the power rule one more time to differentiate (3x + 1)⁴ along with the necessary simplifications.