Problem 3. For each of the following functions f , prove that f is differentiable at any pointa in the domain of f , and find f ′(a).(a) f : R → R, f (x) = x4.(b) f : (0, ∞) → R, f (x) = 1x .(c) f : (0, ∞) → R, f (x) = √x

Problem 5. Let f : R → R be a function, and suppose that f is differentiable at a for anya ∈ R.(a) Prove that for any constant c ∈ R, the function g : R → R, g(x) = f (x) + c isdifferentiable at any a ∈ R with g′(a) = f ′(a).(b) Prove that for any constant c ∈ R, the function g : R → R, g(x) = f (x + c) isdifferentiable at any a ∈ R with g′(a) = f ′(a + c).1

Problem 4. Define the function f : R → R by f (x) = max{0, x}. For each a ∈ R, determineif f is differentiable at a and prove your answer

Problem 2. For each of the following formulas for f (x), find the largest possible set A ⊆ Rso that f : A → R is a function, and prove that f is differentiable at every point a ∈ A.(a) f (x) = sec x = 1cos x .(b) f (x) = csc x = 1sin x .(c) f (x) = tan x = sin xcos x .(d) f (x) = cot x = cos xsin x

9. Let f be the function from R to R defined by f (x) = x2. Finda) f −1({1}) b) f −1({x | 0 < x < 1}) c) f −1({x | x > 4})

b) f(x) = 3𝑥2;     𝑑𝑓𝑑𝑥(1)= therefore the function is