Description
Problem 1
(10 points)
Compute the derivative of the following functions directly from the definition
lim
h→0
f(x + h) − f(x)
h
a) f(x) = x
3
. (4 points)
b) f(x) = p
(x). (4 points)
c) f(x) = x. (1 points)
d) f(x) = c with some constant c. (1 points)
Problem 2
(14 points)
Compute the derivatives of the following functions
a) f(x) = x
2
b−3×2 where b is a constant (2 points)
b) g(t) = cos(ωt + φ) + sin(ωt + φ) where ω and φ are constants (2 points)
c) h(s) = cos(s
2 + s) + sin(s/2) (2 points)
d) j(x) = ln(x
a
2
+ x
−a
2
) where a is a constant
Note: You can use (ln x)
0 = 1/x from the lecture (2 points)
e) k(x) = ln(x
a + b
x
) where a and b are constants (2 points)
f) l(x) = x
2
exp (−x
2
) (2 points)
g) m(x) = x
x
2
(2 points)
Note for e) and g): You cannot directly work with something of the form a
x
(with some a)
but only with something of the form e
cx (with some c). Transform the function accordingly
before differentiation.
Calculus and Linear Algebra
Problem 3
(6 points)
Use the definition of the derivative, f
0
(x) = limh→0
f(x+h)−f(x)
h
, to show that the function
f(x) = |x| is not differentiable at x = 0.
