Exercise 5.2.1

Question

Supply proofs for parts (i) and (ii) of Theorem 5.2.4. (addition and scalar multiplication preserve differentiability)

Ulisse Mini · Accepted Answer

Let $f$ and $g$ be differentiable at $c$. $f + g$ is differentiable at $c$ by using the algebraic limit theorem
  $$
  (f + g)' = \lim_{x 	o c} \frac{(f + g)(x) - (f + g)(c)}{x-c} = \lim_{x 	o c} \frac{f(x) - f(c)}{x - c} \lim_{x 	o c} \frac{g(x) - g(c)}{x - c} = f'(c) + g'(c)
  $$
  Likewise for $k \in \mathbf{R}$ we can apply ALT
  $$
  (kf)' = \lim_{x 	o c} \frac{kf(x) - kf(c)}{x-c} = k \lim_{x 	o c} \frac{f(x)-f(c)}{x-c} = k f'(c)
  $$

Exercise 5.2.1

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Exercise 5.2.1

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