## calyne 2 years ago Differentiate the function f(x) = 5root(ln x)

1. across

$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}}$

2. across

I left out the $$5$$, but you know what I mean.

3. TuringTest

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

4. calyne

oh no yeah it's 5th root

5. across

Then all he has to do is change $$g$$ and $$g'$$ a little.

6. TuringTest

oh good, I got scared for a minute :)

7. calyne

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

8. Will!

$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}$ ${df \over dx} = {df \over du}{du \over dx} = {1 \over x} {{1 \over 5}(\ln x)^{-4/5}} = {1 \over 5x(lnx)^{4 \over 5}}$