Final answer:
The derivative of the function f(w) = w⁷/8 + 7w is f'(w) = 7/8*w^(-1/8) + 7. The domain of f'(w) is all real numbers except for w = 0, so w ∈ ( - ∞, 0) ∪ (0, ∞).
Step-by-step explanation:
To find the derivative of the function f(w) = w⁷/8 + 7w, we use the power rule of differentiation. The power rule states that if f(w) = w^n, then f'(w) = n*w^(n-1). Applying the power rule to both terms in the function:
- For the first term, w⁷/8, the derivative is 7/8*w^(7/8 - 1) = 7/8*w^(-1/8).
- For the second term, 7w, the derivative is 7, because the exponent of w is 1, and 1 * 7 = 7.
Combining both terms, the derivative of the function is f'(w) = 7/8*w^(-1/8) + 7. The domain of f'(w) is all real numbers except for w = 0 because w^(-1/8) is undefined at w = 0. Therefore, the domain of f'(w) is w ∈ ( - ∞, 0) ∪ (0, ∞).