Differentiating Standard Functions: $x^n$ (for any rational $n$), $\sin x$, $\cos x$, $\tan x$, $e^x$, $\ln x$

Introduction

Key Concepts

Understanding Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes concerning its input variable. Mathematically, the derivative of a function $f(x)$ is denoted as $f'(x)$ or $\frac{df}{dx}$. The derivative provides valuable insights into the behavior of functions, including their increasing or decreasing trends and points of maxima or minima.

Power Functions: $x^n$

The derivative of a power function $f(x) = x^n$, where $n$ is any rational number, is given by:

Example: If $f(x) = x^{\frac{3}{2}}$, then:

Trigonometric Functions

• Sine Function ($\sin x$):

  The derivative of $f(x) = \sin x$ is:

  $$f'(x) = \cos x$$

  Example: If $f(x) = \sin(2x)$, using the chain rule:

  $$f'(x) = 2\cos(2x)$$
• Cosine Function ($\cos x$):

  The derivative of $f(x) = \cos x$ is:

  $$f'(x) = -\sin x$$

  Example: If $f(x) = \cos(3x)$, then:

  $$f'(x) = -3\sin(3x)$$
• Tangent Function ($\tan x$):

  The derivative of $f(x) = \tan x$ is:

  $$f'(x) = \sec^2 x$$

  Example: If $f(x) = \tan(4x)$, then:

  $$f'(x) = 4\sec^2(4x)$$

Exponential and Logarithmic Functions

• Exponential Function ($e^x$):

  The derivative of $f(x) = e^x$ is:

  $$f'(x) = e^x$$

  Example: If $f(x) = e^{5x}$, then:

  $$f'(x) = 5e^{5x}$$
• Natural Logarithm ($\ln x$):

  The derivative of $f(x) = \ln x$ is:

  $$f'(x) = \frac{1}{x}$$

  Example: If $f(x) = \ln(6x)$, using the chain rule:

  $$f'(x) = \frac{6}{6x} = \frac{1}{x}$$

Rules of Differentiation

• Sum Rule: The derivative of a sum is the sum of the derivatives. $$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$
• Difference Rule: The derivative of a difference is the difference of the derivatives. $$\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$$
• Product Rule: For two functions $f(x)$ and $g(x)$: $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$
• Quotient Rule: For two functions $f(x)$ and $g(x)$: $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$
• Chain Rule: For a composite function $f(g(x))$: $$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$$

Applications of Differentiation

Differentiation is widely used in various fields such as physics for motion analysis, economics for optimizing profit, and biology for modeling population growth. Understanding how to differentiate standard functions equips students with the tools to solve real-world problems involving rates of change.

Advanced Concepts

In-depth Theoretical Explanations

The differentiation rules for standard functions are derived from the fundamental principles of limits and the definition of the derivative. For instance, the derivative of $f(x) = x^n$ is derived using the limit definition:

Applying the binomial theorem and simplifying leads to:

This foundational approach ensures the robustness of differentiation rules across various functions.

Complex Problem-Solving

Consider the function $f(x) = \frac{e^{2x} \sin x}{x^3}$. To find $f'(x)$, we apply the quotient rule in conjunction with the product rule:

1. Let $u = e^{2x} \sin x$, then $u' = 2e^{2x} \sin x + e^{2x} \cos x$.
2. Let $v = x^3$, then $v' = 3x^2$.
3. Applying the quotient rule:

Simplifying the expression yields:

Applications in Physics and Engineering

In physics, differentiation of trigonometric functions is pivotal in studying oscillatory motions, such as pendulums and springs. For example, the position of a pendulum can be modeled as $f(t) = A \sin(\omega t + \phi)$, where differentiation provides velocity and acceleration:

In engineering, understanding the rate of change of signals can aid in designing control systems and analyzing circuit behaviors.

Interdisciplinary Connections

Differentiation extends beyond pure mathematics, finding relevance in economics for optimizing cost and revenue functions, biology for modeling population dynamics, and even computer science in algorithms optimization. For instance, in economics, the derivative of the cost function $C(x)$ with respect to production level $x$ provides the marginal cost:

Such interdisciplinary applications highlight the versatility and importance of mastering differentiation techniques.

Comparison Table

Summary and Key Takeaways