Final answer:
The derivative of f(x) = ln(x)/xµ is found by applying the quotient rule. After simplification, the derivative, or f'(x), is (1 - 5ln(x))/x⁶.
Step-by-step explanation:
To solve for the derivative of the function f(x) = ln(x)/x⁵, we apply the quotient rule of differentiation. The quotient rule states that the derivative of a function that is the quotient of two other functions (u/v) can be found using the formula (u'v - uv')/v². Here, our u is ln(x) and v is x⁵. The derivative of ln(x) with respect to x is 1/x, and the derivative of x⁵ with respect to x is 5x⁴. Plugging these into the formula, we get:
f'(x) = ((1/x)*x⁵) - (ln(x)*5x⁴))/(x⁵)²
Simplifying, we obtain:
f'(x) = (x⁴ - 5x⁴ln(x))/(x¹⁰)
Further simplification yields:
f'(x) = (1 - 5ln(x))/x⁶