Final answer:
The derivative of the function f(x) = 4√6 x⁸ + 7x⁵ is f'(x) = 32√6 x⁷ + 35x^4, applying the power rule for differentiation and keeping the constant factors intact.
Step-by-step explanation:
To differentiate the function f(x) = 4√6 x⁸ + 7x⁵, we will apply the power rule. The power rule states that if f(x) = x^n, then f'(x) = nx^{n-1}. This works for constants multiplied by powers of x, as well as for fractional exponents that represent roots.
The derivative of f(x) with respect to x is f'(x) = 32√6 x⁷ + 35x^4. For the first term, 4 is a constant factor, so the derivative is 4 times the derivative of x⁸, which is 8x^7. Since √6 is also a constant, it is multiplied as is. The second term 7x⁵ differentiates to 35x^4 by the same rule. Our final answer does not contain fractional or negative exponents.