How do you differentiate f(x)=e^sqrt(1/x)f(x)=e1x using the chain rule.?

2 Answers
Oct 7, 2016

Please follow the instructions below...

Explanation:

y=e^uy=eu

Use part of the chain rule...

:. (dy)/(du)=e^u=e^(sqrt(1/x))

u=sqrt(1/x)

u=(1/x)^(1/2)

u^2=1/x

ln(u^2)=ln(1/x)

2ln(u)=ln1-lnx

2lnu=-lnx

Now use implicit differentiation...

2/u*(du)/(dx)=-1/x

u/2*2/u*(du)/(dx)=-1/x*u/2

(du)/(dx)=-1/(2x)*sqrt(1/x)

Now use the chain rule...

(dy)/(du)*(du)/(dx)=-1/(2x)*e^(sqrt(1/x))*sqrt(1/x)=(dy)/(dx)

Now, you can simplify the final result...

-1/(2x)*e^(sqrt(1/x))*sqrt(1/x)

=-1/(2x)*e^(sqrt(1/x))*sqrt(1)/sqrt(x)*sqrt(x)/sqrt(x)

=-1/(2x)*e^(sqrt(1/x))*sqrt(x)/x

=-1/(2x^2)*e^(sqrt(1/x))*sqrt(x)

Still looks quite ugly, but this is the result you're looking for. It is what it is.

Oct 7, 2016

e^(sqrt(1/x))/(2x^(3/2)

Explanation:

differentiate using the color(blue)"chain rule"

color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|))).... (A)

let u=sqrt(1/x)=(1/x)^(1/2)=1/x^(1/2)=x^(-1/2)

differentiate using the color(blue)"power rule"

rArr(du)/(dx)=-1/2x^(-3/2)

Now y=f(x)=e^urArr(dy)/(du)=e^u

substitute results for (dy)/(du)" and " (du)/(dx) into (A) changing u back to x.

#rArrdy/dx=e^u.-1/2x^(-3/2)=-1/2x^(-3/2)e^(sqrt(1/x))#

rArrdy/dx=(e^(sqrt(1/x)))/(2x^(3/2)