1.(a) Differentiate E(t). Take into account that M is symmetric.
(b) Differentiate p(t).
(c) Compute the derivative of phi.
(d) Differentiate L(t). Take into account that A is antisymmetric.

2.(a) Use product rule for matrix valued functions.
(b) Chain rule?
(c) What is the determinant of a diagonal matrix? How do we invert a diagonal matrix?
(d) Note that A^(-1)A=I, and differentiate both sides with respect to t.

3. f(t)+g(t) can be written as the composition of (f(t),g(t)) and x+y.

4. Use Caratheodory’s condition.

5. Direct computation involving the chain rule.

6. Substitute u=xy, and try so solve for z=z(u). 
The intermediate value theorem and monotonicity may be useful. 
For the differentiability question, use the implicit function theorem.

7. Implicit function theorem. Definition of derivative yields a linear approximation of z=z(x,y).

8. Implicit function theorem. Determinant of a 2x2 matrix.