Differentiation

Question

Two variables x and y are related by y= ln(2+e^-x) for x >0. Given that x is increasing at a rate of 3 units per second, find, in terms of e, the rate of change of y when x=1.

 

Ans: -3/2e+1

 

Thank you!

Answer 1

Finding rates is taking the derivative with respect to time....dy/dt and dx/dt.

 

y=ln(2+e^-2x)

dy/dt = 1/( 2 + e^(-x) ) *(-e ^ (-x) ) dx/dt

 

you know dx/dt = 3 and you're looking for when x =1, so, 

 

dy/dt = 1/ ( 2 + e^-1) * (-e^-1) * 3

dy/dt = 1/ (2+e^-1) * -1/e^1 * 3

dy/dt = -3/( 2e^1 +1)

 

This is the same as your answer.