Verified answer

For the first term, you need to use the product rule and the chain rule. For the second term, you just use the chain rule.

(t^2 * e^-t + (ln t)^2)'

= (t^2)' ( e^-t) + (t^2) (e^-t)' + 2(ln t)^(2-1) * (ln t)'

= 2t (e^-t) + (t^2)(-e^-t) + 2(ln t)*(1/t)

Simplify if your teacher says it is necessary.

For the first term, from the chain rule,

d/dt() = 2*t*e^(-t) + (t^2)(-e^(-t))

For the second term, again from the chain rule,

d/dt() = 2*(ln t)*(1/t)

= (2/t)(ln t)

Add them together since they are a sum in the original function, and get

d/dt() = 2*t*e^(-t) + (t^2)(-e^(-t)) + (2/t)(ln t)

d/dt() = (e^-t)*(2t-t^2) + (2/t)*ln(t)