Differential in a initial value problem

[swooshfactory]

I am trying to solve a differential equation using an integrating factor and the term I am trying to differentiate is (3*t+2*e^t)*e^(-3t/2) .

I have tried simplifying it into these functions (terms, expressions):

[(3*t)*e^(-3t/2)+2*(e^t)(e^(-3t/2))

=3t(e^(-3t/2))+2(e^(-t/2))

=e^(-t/2)(3t(e^-t)+2)


I'm not sure if my work is correct, but the term at the top is right (I'm pretty sure), and I cannot figure out how to differentiate this term. Any suggestions for u-substitutions or other methods I could use to get this derivative would be a big help. Thank you.

 

[NoMoreExams]

If I read this correctly you are trying to differentiate:

$$3t e^{-\frac{3t}{2}} + 2e^{-\frac{t}{2}}$$

For the first part, do you know the product rule? If you don't look it up.

For the first and second part, do you know what the derivative of $$e^{f(x)}$$ is? If you don't look it up as well or use implicit differentiation to figure it out.