The correct derivative of the function f(x) = ln(x)^7 is found using the chain rule and properties of logarithms, resulting in f'(x) = 7/x.
The student is asking for the derivative of the function f(x) = ln(x)7. Using the chain rule and the properties of logarithms, the correct process involves taking the natural logarithm of x raised to the power of 7, which is the product of 7 and ln(x), and then differentiating it with respect to x. The chain rule tells us to differentiate the outer function (in this case, the natural logarithm), and then multiply it by the derivative of the inner function (the x inside the ln).
The derivative of ln(x) is 1/x, and so the derivative of ln(x)7 is 7 times the derivative of ln(x), which gives us 7/x. Therefore, we multiply 7 by 1/x, resulting in the final answer f'(x) = 7/x.
Thus, the correct answer is f'(x) = 7/x.