Final answer:
The missing expression in the derivative d/dx(ln(x^8 + 5)) is 8x^7, making the complete derivative \( \frac{8x^7}{x^8 + 5} \).
Step-by-step explanation:
The question asks for the derivative of the function ln(x8 + 5). To find this derivative, we apply the chain rule which tells us to first differentiate the outer function (in this case the natural logarithm), then multiply by the derivative of the inner function (x8 + 5 in this instance).
Using the chain rule, the derivative of ln(x8 + 5) with respect to x is \( \frac{1}{x8 + 5} \cdot \frac{d}{dx}(x8 + 5) \). The derivative of x8 + 5 with respect to x is simply 8x7, as the derivative of 5, a constant, is 0. Thus, the correct expression for the derivative is \( \frac{8x7}{x8 + 5} \).