Higher Order Derivatives and Derivative of Trigonometric Functions

Tags: #◇Computer Science #◇CSMATH1 #◇Conceptual

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◇Overview

Higher order derivatives refer to the process of taking the derivative of a function multiple times. The derivative of a derivative is called the second derivative, and this pattern continues for higher orders. This section also covers the differentiation of trigonometric functions, which follow specific rules and are essential for solving problems involving periodic functions.

◇Key Definitions

First Derivative ($f'(x)$ or $\frac{dy}{dx}$): Represents the instantaneous rate of change of a function. Second Derivative ($f''(x)$ or $\frac{d^2y}{dx^2}$): Represents the rate of change of the first derivative. Higher Order Derivatives: Derivatives beyond the first, denoted as $f^{(n)}(x)$ for the $n$-th derivative.

◇Theoretical Framework

Derivatives of Trigonometric Functions

The fundamental derivatives of trigonometric functions are: $\frac{d}{dx} (\sin x) = \cos x$ $\frac{d}{dx} (\cos x) = -\sin x$ $\frac{d}{dx} (\tan x) = \sec^2 x$ $\frac{d}{dx} (\cot x) = -\csc^2 x$ $\frac{d}{dx} (\sec x) = \sec x \tan x$ $\frac{d}{dx} (\csc x) = -\csc x \cot x$

Notation for Higher Order Derivatives 1st Derivative: $y'$, $f'(x)$, $\frac{dy}{dx}$ 2nd Derivative: $y''$, $f''(x)$, $\frac{d^2y}{dx^2}$ 3rd Derivative: $y'''$, $f'''(x)$, $\frac{d^3y}{dx^3}$ $n$-th Derivative: $y^{(n)}$, $f^{(n)}(x)$, $\frac{d^ny}{dx^n}$

◇References & Navigation

• Previous Topic: ◇Differentiation Rules
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• Related Notes: ◇Differentiation Rules, ◇The Tangent Line and the Derivative