f(x)= 5ln(x)/x^7

Is there a question here?

Derivative, my apologies.

so y= f(x) then

x^7*y=5ln(x)

7x^6*y+y'x^7=5/x
solve for y'

a bit of algebra will be required. I will do it onscreen but check it, easy to make an error on a keyboard.

y'= 1/x^7 * (5/x -7x^6*5lnx/x^7)
= 1/x^7 * (5/x-35lnx/x) check it.

To find the domain of the function f(x) = 5ln(x)/x^7, we need to consider two restrictions: the domain of the natural logarithm function and any potential division by zero.

1. Domain of the natural logarithm function (ln(x)):
The natural logarithm function, ln(x), is defined for positive real numbers only. Therefore, the argument of the natural logarithm, x, must be greater than 0, i.e., x > 0.

2. Division by zero:
In the denominator, there is an x^7 term. When x = 0, the function will have a division by zero error. Therefore, x ≠ 0.

Combining these two restrictions, the domain of the function f(x) = 5ln(x)/x^7 is x > 0 and x ≠ 0. In interval notation, the domain can be written as (0, ∞).