`f(x) = x^2*e^(-3x)`
1. Ableitung?

`f´(x) = 2x*e^(-3x) + x^2*(-3)*e^(-3x)`
   `= x*e^(-3x)*(2-3x)`
 
`u(x) = x^2`      `u´(x) = 2x`
`v(x) = e^(-3x)`    `v´(x) = -3*e^(-3x)`

`f(x) = x^2*sin(-3x)`
1. Ableitung?

`f´(x) = 2x*sin(-3x)+x^2*(-3)*cos(-3x)`
   `= x*(2sin(-3x) - 3x*cos(-3x))`
 
`u(x) = x^2`      `u´(x) = 2x`
`v(x) = sin(-3x)`    `v´(x) = -3*cos(-3x)`

`f(x) = (x+e^x)^2`
1. Ableitung?

`f´(x) = 2*(x+e^x)*(1+e^x)`

`f(x) = (u(x))/(v(x))`
1. Ableitung?

`f(x) = u(x)*(v(x))^-1`
 
`f´(x) = u´(x) * (v(x))^-1 + u(x)*(-(v(x))^-2)*v´(x)`
   `= (u´(x))/(v(x)) - (u(x)*v´(x))/(v(x))^2`
   `= (u´(x)*v(x) - u(x)*v´(x))/(v(x))^2`   (Quotientenregel)

`f(x) = (x+1)/e^x`
1. Ableitung?

`f(x) = (x+1)*e^(-x)`
 
`f´(x) = 1*e^(-x) + (x+1)*(-e^(-x))`
 
`u(x) = x+1`        `u´(x) = 1`
`v(x) = e^(-x)`        `v´(x) = -e^(-x)`

`f(x) = u(x)*(v(x))^2`
1. Ableitung?

`f´(x) = u´(x)*(v(x))^2 + u(x)*2v(x)*v´(x)`
   `= v(x)[u´(x)*v(x)+2u(x)*v´(x)]`

`f_a(t) = a(t-15)e^(-0,01t) + a*15`
1. Ableitung?

`f_a´(t) = a*e^(-0,01t)+a(t-15)*(-0,01e^(-0,01t))`
   `= a*e^(-0,01t)*(1-0,01t+0,15)`
 
`u(t) = a(t-15)= a*t-a*15`        `u´(t) = a`
`v(t) = e^(-0,01t)`        `v´(t) = -0,01e^(-0,01t)`

`f(x) = 1/(1+x^2)`
1. Ableitung?

`f(x) = (1+x^2)^(-1)`
 
`f´(x) = -(1+x^2)^(-2)*2x`
   `= (-2x)/(1+x^2)^2`

`f_t(x) = sin(tx) + 2`
1. Ableitung?

`f_t´(x) = t*cos(tx)`

`f_k(x) = k*e^(kx)-8`
1. Ableitung?

`f_k´(x) = k^2*e^(kx)`