how do I differentiate this natural log?

Line 2 to line 3, you need parentheses:

|ln(a·b)|c - ln(a·b)|
|:-|:-|
|ln(a)+ln(b)|c - (ln(a)+ln(b))|

Line 3 to line 4, you need the chain rule:

|f(x)|ln(x)|ln(7x)|ln(-x)
|:-|:-|:-|:-|
|f'(x)|1/x|7·(1/7x)|-1·(1/-x)|

Line 6 to line 7, you didn't change the numerator of the left fraction, you only multiplied the denominator

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