Calculus Concepts: Derivatives

1. Tangent and Velocity

Intuition

The concept of a derivative originates from the need to find the slope of a tangent line to a curve at a point. In physics, this translates to determining the instantaneous velocity of an object.

The slope of a secant line connecting two points on a curve gives an average rate of change. As the two points move closer, the secant line approaches the tangent line, and the slope approaches the derivative.

Mathematical Definition

The derivative of a function \( f(x) \) at a point \( x = a \) is given by:

\[ \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]

Example

Find the derivative of \( f(x) = x^2 \) at \( x = 2 \).

\[ f'(2) = \lim_{h \to 0} \frac{(2 + h)^2 - 2^2}{h} = \lim_{h \to 0} \frac{4 + 4h + h^2 - 4}{h} = \lim_{h \to 0} (4 + h) = 4 \]

Python Visualization

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(0, 4, 100)
f = x**2

def tangent_line(x, a):
    slope = 2 * a
    return slope * (x - a) + a**2

x_tangent = 2
y_tangent = tangent_line(x, x_tangent)

plt.plot(x, f, label='f(x) = x^2')
plt.plot(x, y_tangent, '--', label=f'Tangent at x={x_tangent}')
plt.scatter([x_tangent], [x_tangent**2], color='red')
plt.legend()
plt.title('Tangent and Velocity')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid()
plt.show()

2. Rate of Change

Intuition

The rate of change measures how a quantity changes with respect to another. For example, speed is the rate of change of distance with respect to time.

Example

The rate of change of \( f(x) = x^3 \) from \( x = 1 \) to \( x = 2 \) is:

\[ \text{Average rate of change} = \frac{f(2) - f(1)}{2 - 1} = \frac{8 - 1}{1} = 7 \]

Python Visualization

x = np.linspace(0, 3, 100)
f = x**3

x1, x2 = 1, 2
y1, y2 = x1**3, x2**3

plt.plot(x, f, label='f(x) = x^3')
plt.plot([x1, x2], [y1, y2], 'o-', label='Secant Line')
plt.legend()
plt.title('Rate of Change')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid()
plt.show()

3. Derivative as a Function

Intuition

The derivative itself can be treated as a function, \( f'(x) \), that gives the slope of the tangent line to \( f(x) \) at any point \( x \).

Example

For \( f(x) = x^3 \),

\[ f'(x) = 3x^2 \]

Python Visualization

x = np.linspace(-2, 2, 100)
f = x**3
f_prime = 3 * x**2

plt.plot(x, f, label='f(x) = x^3')
plt.plot(x, f_prime, label="f'(x) = 3x^2")
plt.legend()
plt.title('Derivative as a Function')
plt.xlabel('x')
plt.ylabel('y')
plt.grid()
plt.show()

4. Review of Derivative

Intuition

Derivatives are foundational to calculus, enabling us to: - Calculate slopes of tangent lines. - Analyze rates of change. - Solve real-world problems in physics, biology, and economics.

Key Rules

Power Rule: \( \frac{d}{dx} [x^n] = nx^{n-1} \)
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: \( \frac{d}{dx} [f(g(x))] = f'(g(x))g'(x) \)

5. Mean Value Theorem

Intuition

If \( f(x) \) is continuous and differentiable on \([a, b]\), there exists a point \( c \) in \((a, b)\) such that:

\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

Example

For \( f(x) = x^2 \) on \([1, 3]\):

\[ \text{Average slope} = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4 \]

The derivative \( f'(x) = 2x \) satisfies \( f'(c) = 4 \) at \( c = 2 \).

Python Visualization

x = np.linspace(0, 4, 100)
f = x**2

x1, x2 = 1, 3
y1, y2 = x1**2, x2**2

def mean_value(x):
    return 4 * (x - 2) + 4

x_mvt = 2
y_mvt = mean_value(x)

plt.plot(x, f, label='f(x) = x^2')
plt.plot([x1, x2], [y1, y2], 'o-', label='Secant Line')
plt.plot(x, y_mvt, '--', label='Tangent at x=2')
plt.legend()
plt.title('Mean Value Theorem')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid()
plt.show()

6. Indeterminate Forms and L'Hopital's Rule

Intuition

Indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) arise in calculus. L'Hopital's Rule provides a method to evaluate such limits:

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}, \text{ if the limit exists.} \]

Example

Evaluate \( \lim_{x \to 0} \frac{\sin x}{x} \).

\[ \text{Using L'Hopital's Rule: } \lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1 \]

Python Visualization

x = np.linspace(-1, 1, 100)
y = np.sin(x) / x

def safe_division(x):
    return np.where(x == 0, 1, np.sin(x) / x)

y_safe = safe_division(x)

plt.plot(x, y_safe, label='sin(x)/x')
plt.axhline(1, color='red', linestyle='--', label='y=1')
plt.legend()
plt.title("L'Hopital's Rule")
plt.xlabel('x')
plt.ylabel('y')
plt.grid()
plt.show()