Exercises

1. Use the Chain Rule and the rule for differentiating the square root function to calculate the derivative of each of the following functions.

   `sqrt(2t+3)`

   `sqrt(2t^2+3)`

   `sqrt(t^3-t)`

2. Use the Chain Rule and the Product Rule (if needed) to calculate the derivative of each of the following functions.
   

   `e^(4t-6)` `e^(t^2)` `e^(t+4)` `t e^(4t-6)`

   `t e^(t^2)` `t e^(t+4)` `t^2e^(4t-6)` `t^2e^(t^2)`

   `t^2e^(t+4)` `sqrt(t ) e^(4t-6)` `sqrt(t ) e^(t^2)` `sqrt(t ) e^(t+4)`

3. For each of the following functions, find `(dy)/(dx)`.

   `y=xe^(-x)`

   `y=e^(x^2+x)`

   `y=(2x^3+2x)^2`

4. For each of the following functions, find `(dy)/(dx)`.

   `y=sqrt(x^5+x^2+1)` `y=u^2+u`, where `u=sqrt(x^2+1)` `y=v^2+v`, where `v=sqrt(u^2+1)` and `u=x^3-x`

5. If `z=y^3+y+2` and `y=3x+1`, find `(dz)/(dx)`.
6. Suppose `f` and `g` are function such that `f'text[(]2text[)]=4`, `g'text[(]2text[)]=-3`, `ftext[(]2text[)]=-1`, `gtext[(]2text[)]=1`, `f'text[(]1text[)]=2`, and `g'text[(]-1text[)]=5`. For each of the following functions, find the value of the derivative at `x=2`.
   

   `stext[(]xtext[)]=ftext[(]xtext[)]+gtext[(]xtext[)]` `ptext[(]xtext[)]=ftext[(]xtext[)]gtext[(]xtext[)]`

   `htext[(]xtext[)]=ftext[(]gtext[(]xtext[))]` `ktext[(]xtext[)]=gtext[(]ftext[(]xtext[))]`

   
   This exercise is adapted from Calculus Problems for a New Century, edited by Robert Fraga, MAA Notes No. 28, 1993.