Differentiation

Definition of Differentiation

Differentiation is a fundamental concept in calculus used to determine the rate at which a quantity changes. It helps find the derivative of a function, representing the slope of the tangent to the curve at any point.

This formula provides the instantaneous rate of change of $f(x)$ at $x$.

Rules of Differentiation

Differentiation is governed by specific rules that simplify the computation of derivatives:

• Power Rule:
  $$\frac{d}{dx} [x^n] = n \cdot x^{n-1}$$
  Example: $\frac{d}{dx} [x^3] = 3x^2$
• Sum Rule:
  $$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$$
• Product Rule:
  $$\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$
• Quotient Rule:
  $$\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$
• Chain Rule:
  $$\frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x)$$

Examples of Differentiation

Example 1:

Differentiate $f(x) = 3x^4 + 5x^2 - 7$.

Solution:

So, the derivative is $12x^3 + 10x$.

Example 2:

Differentiate $f(x) = (x^2 + 1)(x^3 - x)$ using the product rule.

Solution:

The derivative is $5x^4 - 3x^2 - 1$.

Applications of Differentiation

Differentiation has extensive applications in various fields:

• Physics: Calculating velocity and acceleration from position-time functions.
• Economics: Determining marginal cost and marginal revenue functions.
• Engineering: Analyzing stress and strain relationships in materials.
• Biology: Modeling growth rates of populations or cells.

Practice Question

Problem:

Differentiate $f(x) = \frac{x^2 + 1}{x + 2}$.

Solution:

1. Use the quotient rule:
   $$\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$
2. Here, $f(x) = x^2 + 1$ and $g(x) = x + 2$:
   $$f'(x) = 2x, \; g'(x) = 1$$
3. Substitute into the formula:
   $$\frac{d}{dx} \left[\frac{x^2 + 1}{x + 2}\right] = \frac{(2x)(x+2) - (x^2+1)(1)}{(x+2)^2}$$
   $$= \frac{2x^2 + 4x - x^2 - 1}{(x+2)^2}$$
   $$= \frac{x^2 + 4x - 1}{(x+2)^2}$$

The derivative is $\frac{x^2 + 4x - 1}{(x+2)^2}$.