notes-archive/docs/math/real-analysis/6-derivatives.md

Chapter 6 - Derivatives

Section 6.1 - The Derivative

Definition. Let I \subseteq \mathbb{R} be an interval f: I \rightarrow \mathbb{R} a function, and c \in I. Then, the derivative of f at $c$ is

f'(c) = \lim_{x \rightarrow c} \frac{f(x)-f(c)}{x-c}

provided that the limit exists. Thus, we obtain f'(x), or the derivative of $f$, with a domain of all points c \in I where the limit exists.

Theorem. If f: I \rightarrow \mathbb{R} is differentiable at point c, it is continuous at point c.

Theorem. Let f, g: I \rightarrow \mathbb{R} be differentiable at c \in I. Then,

• (f+g)'(c) = f'(c) + g'(c)
• (fg)'(c) = f'(c)g(c) + f(c)g'(c)
• (\frac{f}{g})'(c) = \frac{f'(c)g(c) - f(c)g'(c)}{(g(c))^2}

Theorem. Let f: I \rightarrow \mathbb{R} and g: J \rightarrow \mathbb{R}, with f(I) \subseteq J. Then, if f is differentiable at c \in I and g is differentiable att f(c) \in J, then f \circ g is differentiable at c, and

(g \circ f)'(c) = g'(f(c))f'(c)

Theorem. Suppose f: I \rightarrow \mathbb{R} is a one-to-one function on some interval I. Then, with J = f(I) and f^{-1}: J \rightarrow \mathbb{R}, for all x \in I,

f^{-1}(f(x)) = x

Lemma. f is continuous on some interval I if and only if it is monotonic on said interval.

Theorem. Suppose f: I \rightarrow \mathbb{R} is a one-to-one function on some interval I. Then, with J = f(I) and f^{-1}: J \rightarrow \mathbb{R}, if f is continuous on I and I is an interval, then f(I) is an interval, and f^{-1} is continuous on J.

Theorem. Suppose f: I \rightarrow \mathbb{R} is differentiable at some c \in I. Then, with J = f(I) and f^{-1}: J \rightarrow \mathbb{R}, if f is differentiable at c and f'(c) \neq 0, then f^{-1} is differentiable at d = f(c), and

(f^{-1})(d) = \frac{1}{f'(c)}

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Definition. Let I \subseteq \mathbb{R} be an interval and let f: I \rightarrow \mathbb{R}. Then, f has a relative maximum (or local maximum) at some point c \in I if there exists some $\delta$-neighborhood V_\delta(c) such that for all x \in V_\delta(C) \cup I, then f(x) \leq f(c). Relative minima are defined similarly.

Theorem. Let I be an interval and f: I \rightarrow \mathbb{R}. Then, if f has a relative extremum at an interior point c \in I, and if f'(c) exists, then f'(c) = 0.

Corollary. Suppose f: [a, b] \rightarrow \mathbb{R} and g: [a, b] \rightarrow \mathbb{R} are both continuous on [a, b] and differentiable on (a, b) with a \neq b. Then,

• Rolle's Theorem. If f(a) = f(b), then there exists at least one point c \in (a, b) with f'(c) = 0.
• If f(a) = g(a) and f(b) = g(b), then there exists aat least one point c \in (a, b) such that f'(c) = g'(c).

Theorem. Mean Value Theorem. Let f be continuous on [a, b] and differentiable on (a, b), with a \neq b. Then, there exists at least one c \in (a, b) with

f(b) - f(a) = f'(c)(b - a)

Theorem. Suppose f: I \rightarrow \mathbb{R} is differentiable on I. Then,

• If f'(x) > 0 for all $x \in $I, then f is strictly increasing on I.
• If f'(x) = 0 for all $x \in $I, then f is constant on I.
• If f'(x) <> 0 for all $x \in $I, then f is strictly decreasing on I.

Theorem. Cauchy Mean Value Theorem. Let f, g be continuous on [a, b] and differentiable on (a, b). If g'(x) \neq 0 for all x \in (a, b), and a \neq b, then there exists at least one point c \in (a, b) such that

\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(c)}{g'(c)}