How do you differentiate $f (x) = \sqrt{\frac{ln (\frac{1}{x})}{x^{2}}}$ $f(x)= sqrt (ln(1/x)/x^2$ ?

Question

1986 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-12-29T19:32:47

Using chain and quotient rule.

Chain rule states that $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}$ $(dy)/(dx)=(dy)/(du)(du)/(dx)$
Quotient rule states that for $y = \frac{f (x)}{g (x)}$ $y=f(x)/g(x)$ , $y'=(f'g-fg')/g^2$

We can go renaming parts of the expression in order to make up the chain.

$f(x)=sqrt(u)$
$u=ln(v)/x^2$
$v=1/x$

$(df(x))=(dx)=1/(2u^(1/2))*((1/v)(-1/x^2)(x^2)-ln(1/x)2x)/x^4$

Substituting $v$ , then $u$ and aggregating what's possible, so far

$(df(x))/(dx)=1/(2u^(1/2))*((1/(1/x))(-1/cancel(x^2))(cancel(x^2))-ln(1/x)2x)/x^4$

$(df(x))/(dx)=1/(2((ln(1/x))/x^2)^(1/2))*((x)(-1/cancel(x^2))(cancel(x^2))-ln(1/x)2x)/x^4$

$(df(x))/(dx)=(-x-2xln(1/x))/(2x^4sqrt((ln(1/x))/x^2))=color(green)((-1-2ln(1/x))/(2x^3sqrt((ln(1/x))/x^2)))$

How do you differentiate f(x)= sqrt (ln(1/x)/x^2f(x)=√ln(1x)x2f(x)= sqrt (ln(1/x)/x^2?