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Derivatives and Critical Points Analysis

Derivatives

First Derivative $f'(x)$ : Reflects the slope or rate of change of $f(x)$ at any point $x$ .
Second Derivative $f''(x)$ : Reflects the curvature or concavity of $f(x)$ at any point $x$ .
Higher Order Derivatives: The $n$ -th derivative $f^{(n)}(x)$ represents the rate of change of the $(n-1)$ -th derivative.

Critical Points

A point $x = c$ is a critical point if $f'(c) = 0$ or if $f'(c)$ does not exist.
Local Minimum: $f'(x) = 0$ and $f''(x) > 0$ .
Local Maximum: $f'(x) = 0$ and $f''(x) < 0$ .
Inflection Point: A point where $f''(x) = 0$ indicating a potential change in concavity.

Newton's Method

An iterative algorithm to find successively better approximations to the roots of a real-valued function.
Given by the formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

Implicit Differentiation

Used for functions $y = g(x)$ that are not explicitly solved for $y$ . The derivative $dy/dx$ is found using the chain rule and other differentiation rules.

Increasing and Decreasing Intervals

Increasing: $f'(x) > 0$ implies that $f(x)$ is increasing on that interval.
Decreasing: $f'(x) < 0$ implies that $f(x)$ is decreasing on that interval.

Concavity and Points of Inflection

Concave Up: If $f''(x) > 0$ for all $x$ in an interval, $f(x)$ is concave up on that interval.
Concave Down: If $f''(x) < 0$ for all $x$ in an interval, $f(x)$ is concave down on that interval.
Inflection Point: If the sign of $f''(x)$ changes at $x = c$ , then $c$ is an inflection point.

References

Derivatives
Critical Points
Newton's Method
Implicit Differentiation
Increasing and Decreasing Intervals
Concavity and Points of Inflection
References