Final answer:
To find the derivative of the function f(x) = (x³ + 3x + 5)⁴, we apply the chain rule. The derivative is f'(x) = 4(x³ + 3x + 5)³ * (3x² + 3).
Step-by-step explanation:
To find f'(x) for the function f(x) = (x³ + 3x + 5)⁴, we use the chain rule of differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Here, let u = x³ + 3x + 5. Then f(x) = u⁴. We must first find the derivative u' which is the derivative of x³ + 3x + 5. Using basic differentiation rules, u' = 3x² + 3.
Now, we differentiate the outer function, which is u⁴. The derivative of u^n with respect to u is n*u^(n-1). Thus, the derivative of u⁴ with respect to u is 4u³.
Finally, applying the chain rule, we get:
f'(x) = 4u³ * u' = 4(x³ + 3x + 5)³ * (3x² + 3).