Question

Differentiate the function f(x) = 5root(ln x)

Answer 1

\[f(x)=5\sqrt{\ln x}\]\[g(x)=\sqrt{x}\]\[h(x)=\ln x\]\[f(x)=g(h(x))\]\[g'(x)=\frac{1}{2\sqrt{x}}\]\[h'(x)=\frac{1}{x}\]\[f'(x)=g'(h(x))h'(x)=\frac{1}{2x\sqrt{\ln x}}\]

Answer 2

I left out the \(5\), but you know what I mean.

Answer 3

but what if they mean\[\large \sqrt[5]{\ln x}\]?

Answer 4

oh no yeah it's 5th root

Answer 5

Then all he has to do is change \(g\) and \(g'\) a little.

Answer 6

oh good, I got scared for a minute :)

Answer 7

and don't use substitution for the composite functions just go at it

Answer 8

\[f(x) = (\ln|x|)^{1 \over 5}\]\[\ln x = u \rightarrow f(x) = u^{1 \over 5}\]\[{df \over du} = {1 \over 5}u^{-4 \over 5} \]\[{du \over dx} = {1 \over x}\]
\[{df \over dx} = {df \over du}{du \over dx} = {1 \over x} {{1 \over 5}(\ln x)^{-4/5}} = {1 \over 5x(lnx)^{4 \over 5}} \]