Final answer:
To find the derivative f'(x) of the function f(x) = 5 + 7ln(6/x³), we apply the chain rule and power rule of differentiation to get f'(x) = -21/x.
Step-by-step explanation:
To find f'(x) for the function f(x) = 5 + 7ln(6/x³), we'll use the rules of differentiation. First, the derivative of a constant is zero, so the derivative of 5 is 0. For the second term, we will apply the chain rule.
The derivative of ln(u), where u is a function of x, is 1/u times the derivative of u with respect to x.
In this case, u = 6/x³.
The derivative of 6/x³ with respect to x is -18/x´, by using the power rule of differentiation.
Therefore, the derivative of 7ln(6/x³) with respect to x is 7 * (-18/x´) divided by 6/x³ which simplifies to -21/x.
Adding this to the derivative of 5 (which is 0), we get that f'(x) = -21/x.