Introduction to Derivatives

The derivative is one of the two fundamental concepts of calculus (the other being the integral). It measures the rate of change of a function -- how fast something is changing at any given instant. Derivatives are used everywhere: velocity, acceleration, growth rates, and optimization.

What is a Derivative?

The derivative of y = f(x) at a point represents the slope of the tangent line to the curve at that point. It measures the instantaneous rate of change.

Derivative from First Principles

The derivative is defined as a limit:

f'(x) = lim (h-->0) [f(x + h) - f(x)] / h

Example: Find the derivative of f(x) = x² from first principles.

Solution:

• f(x + h) = (x + h)² = x² + 2xh + h²
• f(x + h) - f(x) = 2xh + h²
• [f(x + h) - f(x)] / h = 2x + h
• lim (h-->0) (2x + h) = 2x

So the derivative of x² is 2x. This means at x = 3, the slope is 6; at x = -1, the slope is -2.

Differentiation Rules

Instead of using first principles every time, we use these rules:

Power Rule

d/dx (xⁿ) = n x^(n-1)

Examples:

• d/dx (x³) = 3x²
• d/dx (x⁵) = 5x⁴
• d/dx (x) = 1
• d/dx (constant) = 0

Constant Multiple Rule

d/dx [c f(x)] = c f'(x)

Example: d/dx (5x³) = 5 x 3x² = 15x²

Sum/Difference Rule

d/dx [f(x) + g(x)] = f'(x) + g'(x)

Example: d/dx (x³ + 4x² - 7x + 2) = 3x² + 8x - 7

Worked Examples

Example 1: Differentiate f(x) = 3x⁴ - 2x³ + 5x - 8

f'(x) = 12x³ - 6x² + 5

Example 2: Find the slope of y = x² - 3x + 1 at x = 2.

• y' = 2x - 3
• At x = 2: y' = 2(2) - 3 = 1

Example 3: Differentiate f(x) = 4/x = 4x⁻¹

f'(x) = 4(-1)x⁻² = -4/x²

Applications of Derivatives

1. Velocity and Acceleration: If s(t) is the position of an object at time t:

• Velocity: v(t) = ds/dt = s'(t)
• Acceleration: a(t) = dv/dt = s''(t)

Example: A ball is thrown upward: s(t) = 20t - 5t² (metres, seconds).

• v(t) = 20 - 10t
• v = 0 when t = 2 s (ball reaches maximum height)
• Maximum height: s(2) = 20(2) - 5(4) = 40 - 20 = 20 m

2. Finding maxima/minima: Where f'(x) = 0, the function may have a maximum or minimum.

Nepal Connection

Engineers designing roads through Nepal's hills use derivatives to calculate slope gradients. Economists use derivatives to analyze Nepal's GDP growth rate.

Key Takeaways

• The derivative measures the instantaneous rate of change
• Power rule: d/dx(xⁿ) = nxⁿ⁻¹
• Derivatives of sums/differences are computed term by term
• Velocity is the derivative of position; acceleration is the derivative of velocity