Question

Let \(\Large f(x) = log(1-x)\)

a) Write down all higher derivatives \(f^{(k)}(x)\) for \(k\geq 1\)
in closed form (i.e. as rational functions)

Answer 1

\[\frac{1}{1-x}\]
\[\frac{1}{(1-x)^2}\]
\[\frac{2}{(1-x)^3}\]
\[\frac{6}{(1-x)^4}\]
\[...\]

Answer 2

pattern is clear yes? power in the denominator increases by 1 each time
the numerator may or may not be clear, but one more should suffice
\[\frac{24}{(1-x)^5}=\frac{4\times 6}{(1-x)^5}=\frac{4\times 3\times 2}{(1-x)^5}\]

Answer 3

@satellite73 thank you i was just asking about it, you read my mind :) just numerator i could not figure out .. how we get 1 2 6 i would say 1x2x3 but .. i dont know :)

Answer 4

Now i get it :) thank you...