Final Answer:
The derivative of the given function f(x) = 5x^4^4 - 4x^8^3 is f'(x) = 20x^4^3-1 - 96x^8^2-1.
Step-by-step explanation:
To find the derivative, we apply the power rule of differentiation. For the first term, 5x^4^4, the power rule yields 20x^4^4-1, where we subtract 1 from the exponent and multiply by the original exponent. Similarly, for the second term -4x^8^3, the power rule gives -96x^8^3-1. This results from subtracting 1 from the exponent and multiplying by the original exponent.
Understanding the power rule is fundamental in calculus, as it allows us to find the derivatives of functions with variable exponents efficiently. In this case, applying the power rule to each term of the given function provides the derivative, f'(x) = 20x^4^3-1 - 96x^8^2-1.
In conclusion, differentiating the function involves applying the power rule separately to each term, resulting in the final answer for the derivative. This process is crucial in calculus for analyzing the rate of change of functions and understanding their behavior.