how to use the chain rule with e in the funciton

Question

My teacher confused me on the example where she said y=te^(-t²). Using the chain rule where its the derivative of the inner times the derivative of the outer I don't know how to do this because she said the outer is the and the inner is the exponent but that doesn't make sense. Could you explain what she meant because I have no idea how to break this apart.

Answer 1

If u is a function of t, then (d/dt)(e^u) = e^u(du/dt)

 

y = te^{-t^2} is a product of two functions.

 

By the product rule, y' = (t)'(e^{-t^2}) + t(e^{-t^2})'

 

                                = e^{-t^2} + t(e^{-t^2})(-2t)

 

                                = e^{-t^2}(1 - 2t²)