Problem 1
Compute the derivative of the following functions directly from the definition
a) f(x) = x3.
b) f(x) = p(x).
c) f(x) = x.
d) f(x) = c with some constant c.
Problem 2
Compute the derivatives of the following functions
a) where b is a constant
b) g(t) = cos(ωt + φ) + sin(ωt + φ) where ω and φ are constants
c) h(s) = cos(s2 + s) + sin(s/2)
d) j(x) = ln(xa2 + x−a2) where a is a constant
Note: You can use (lnx)0 = 1/x from the lecture e) k(x) = ln(xa + bx) where a and b are constants (2 points)
f) l(x) = x2 exp(−x2) (2 points)
g) m(x) = xx2 (2 points)
Note for e) and g): You cannot directly work with something of the form ax (with some a) but only with something of the form ecx (with some c). Transform the function accordingly before differentiation.
Problem 3
(6 points)
Use the definition of the derivative, , to show that the function f(x) = |x| is not differentiable at x = 0.
