Notes on Derivatives and Taylor Series Calculations

Overview of Derivatives and Taylor Series

Finding Derivatives

• Objective: To compute the first several derivatives of a given function at a specific point, then evaluate these derivatives.
• Step 1: Compute the first, second, third, fourth, and potentially fifth derivatives of the function. This process is often applied to various functions, including trigonometric, exponential, and polynomial functions.
• Evaluation Point: Derivatives are evaluated at a specified point (e.g., $x = 0$ or $x = 1$), which is crucial for constructing Taylor series.

Example: Evaluation of Derivatives

Let's consider the function $f(x) = 3\text{sin}(x)$ and evaluate its first four derivatives at $x = 0$.

• Function: $f(x) = 3\text{sin}(x)$

• First Derivative:

  • Calculation: $f'(x) = \frac{d}{dx}(3\text{sin}(x)) = 3\text{cos}(x)$
  • Evaluation at zero: $f'(0) = 3\text{cos}(0) = 3 \cdot 1 = 3$

• Second Derivative:

  • Calculation: $f''(x) = \frac{d}{dx}(3\text{cos}(x)) = -3\text{sin}(x)$
  • Evaluation at zero: $f''(0) = -3\text{sin}(0) = -3 \cdot 0 = 0$

• Third Derivative:

  • Calculation: $f'''(x) = \frac{d}{dx}(-3\text{sin}(x)) = -3\text{cos}(x)$
  • Evaluation at zero: $f'''(0) = -3\text{cos}(0) = -3 \cdot 1 = -3$

• Fourth Derivative:

  • Calculation: $f''''(x) = \frac{d}{dx}(-3\text{cos}(x)) = 3\text{sin}(x)$
  • Evaluation at zero: $f''''(0) = 3\text{sin}(0) = 3 \cdot 0 = 0$

• Note: The number of derivatives needed (up to five or more) depends on the desired precision for the function approximation.

Taylor Series Construction

• Definition: A Taylor series approximates a function, $f(x)$, as an infinite sum of terms, where each term is calculated from the values of the function's derivatives at a single point, $a$ (often called the center of the series).

• General form: The Taylor series for $f(x)$ expanded around point $a$ is given by:
  $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \dots$

• Summation Form: This can be concisely written using summation notation as:
  $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$

  • For this discussion, the point of expansion is referred to as $a$ or $c$. For example, if $c = 1$, the terms will involve $(x-1)$ powers.

• Specific Calculation Steps (for constructing terms):

  1. Determine the expansion point, $a$.
  2. Calculate the function value $f(a)$.
  3. Calculate the first derivative $f'(x)$ and evaluate $f'(a)$.
  4. Calculate higher-order derivatives $f''(x), f'''(x), \dots, f^{(n)}(x)$ and evaluate them at $a$.
  5. Substitute these values into the Taylor series formula. For example, a term might look like $\frac{f^{(n)}(a)}{n!}(x-a)^n$.

Example for Taylor Series Calculation

• Given: Taylor series for $f(x) = \text{ln}(x)$ expanded around $c = 1$. The general term for the Taylor series of $\text{ln}(x)$ around $c=1$ is given by: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(x-1)^n = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 - \dots$
  • Note that for $\text{ln}(x)$, $f(1) = \text{ln}(1) = 0$, so the series starts from $n = 1$.

Derivative Notation and Understanding

• Notation: Various notations are used to represent derivatives:
  • Leibniz's notation: $\frac{dy}{dx}$, $\frac{d}{dx}f(x)$, $\frac{d^2y}{dx^2}$ for second derivative.
  • Lagrange's notation (Prime notation): $f'(x)$, $f''(x)$, $f'''(x)$, $f^{(n)}(x)$.
  • Newton's notation (Dot notation): $\dot{y}, \ddot{y}$ (primarily used in physics for derivatives with respect to time).
  • Euler's notation (D-operator notation): $D<em>xf(x)$, $$D