We have:
x=e^sqrtt
y=1/t+2
We manipulate the equations a bit.
=>ln(x)=sqrtt
=>(ln(x))^2=t
=>y-2=1/t
=>1/(y-2)=t
We now have:
(ln(x))^2=1/(y-2)
=>1/(ln(x))^2=y-2
=>(ln(x))^-2+2=y
=>d/dx[(ln(x))^-2+2]=d/dx[y]
=>d/dx[(ln(x))^-2]+d/dx[2]=dy/dx
Power rule:
d/dx[x^n]=nx^(n-1)
d/dx[ln(x)]=1/x
Chain rule:
d/dx[f(g(x))]=f'(g(x))*g'(x)
=>-2(ln(x))^(-2-1)*d/dx[ln(x)]+0*2*x^(0-1)=dy/dx
=>-2(ln(x))^(-3)*1/x+0=dy/dx
=>-2/(ln(x))^(3)*1/x=dy/dx
=>-2/(x(ln(x))^(3))=dy/dx
Substitute by using the fact that e^sqrtt=x
=>-2/(e^sqrtt(ln(e^sqrtt))^(3))=dy/dx
=>-2/(e^sqrtt(t^(1/2))^(3))=dy/dx
=>-2/(e^sqrtt(t^(3/2)))=dy/dx
Replace t with 1.
=>-2/(e^sqrt1(1^(3/2)))=f'(1)
=>-2/(e)=f'(1)