How do you differentiate # f(x)=e^sqrt(1/x)# using the chain rule.?
2 Answers
Please follow the instructions below...
Explanation:
Use part of the chain rule...
Now use implicit differentiation...
Now use the chain rule...
Now, you can simplify the final result...
Still looks quite ugly, but this is the result you're looking for. It is what it is.
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)#