Integrals by Partial Fractions – Rational Function Integration

🔑

Key Formula

\[ dy = f'(x) \, dx = \frac{dy}{dx} \cdot dx \]

🎯 What does this mean?

This formula shows that the differential dy represents the approximate change in y when x changes by a small amount dx, calculated using the derivative as the rate of change multiplier.

📐

Basic Differential Definition

For a function y = f(x), the differential is defined as:

\[ dy = f'(x) \, dx \quad \text{(Differential of y)} \]

\[ dx = \Delta x \quad \text{(Differential of x is just the change in x)} \]

\[ \Delta y \approx dy \quad \text{(Actual change ≈ Differential approximation)} \]

\[ \text{where } \Delta y = f(x + \Delta x) - f(x) \text{ (actual change)} \]

🔗

Basic Differentiation Rules

Fundamental rules for finding differentials of various functions:

\[ dy = y' \, dx \quad \text{(Total differential of a function)} \]

\[ d(Cu) = C \, du \quad \text{(Constant multiple rule)} \]

\[ d(u + v - w) = du + dv - dw \quad \text{(Sum and difference rule)} \]

\[ d(uv) = u \, dv + v \, du \quad \text{(Product rule)} \]

\[ d(uvw) = (vw) \, du + (uw) \, dv + (uv) \, dw \quad \text{(Product rule extension)} \]

\[ d\left(\frac{u}{v}\right) = \frac{v \, du - u \, dv}{v^2} \quad \text{(Quotient rule)} \]

\[ d(u^n) = nu^{n-1} \, du \quad \text{(Power rule)} \]

📊

Trigonometric Function Differentials

Differentials of common trigonometric functions:

\[ d(\sin u) = \cos u \, du \]

\[ d(\cos u) = -\sin u \, du \]

\[ d(\tan u) = \sec^2 u \, du \]

\[ d(\cot u) = -\csc^2 u \, du \]

\[ d(\sec u) = \sec u \tan u \, du \]

\[ d(\csc u) = -\csc u \cot u \, du \]

📈

Exponential and Logarithmic Differentials

Differentials of exponential and logarithmic functions:

\[ d(e^u) = e^u \, du \]

\[ d(a^u) = a^u \ln a \, du \]

\[ d(\ln u) = \frac{1}{u} \, du \]

\[ d(\log_a u) = \frac{1}{u \ln a} \, du \]

\[ d(u^v) = u^v \left[ v \frac{du}{u} + \ln u \, dv \right] \quad \text{(Power of function)} \]

🔄

Inverse Trigonometric Differentials

Differentials of inverse trigonometric functions:

\[ d(\arcsin u) = \frac{1}{\sqrt{1-u^2}} \, du \]

\[ d(\arccos u) = -\frac{1}{\sqrt{1-u^2}} \, du \]

\[ d(\arctan u) = \frac{1}{1+u^2} \, du \]

\[ d(\text{arccot } u) = -\frac{1}{1+u^2} \, du \]

\[ d(\text{arcsec } u) = \frac{1}{|u|\sqrt{u^2-1}} \, du \]

\[ d(\text{arccsc } u) = -\frac{1}{|u|\sqrt{u^2-1}} \, du \]

📊

Linear Approximation Formula

Using differentials to approximate function values:

\[ f(x + dx) \approx f(x) + f'(x) \, dx \]

\[ f(a + h) \approx f(a) + f'(a) \cdot h \quad \text{(Linearization at point a)} \]

\[ L(x) = f(a) + f'(a)(x - a) \quad \text{(Tangent line approximation)} \]

\[ \text{Example: } \sqrt{9.1} \approx 3 + \frac{1}{6}(0.1) = 3.0167 \]

🔄

Chain Rule for Differentials

Differentials of composite functions:

\[ \text{If } y = f(u) \text{ and } u = g(x), \text{ then } dy = f'(u) \, du = f'(g(x)) \cdot g'(x) \, dx \]

\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \quad \text{(Chain rule form)} \]

\[ \text{Example: If } y = (x^2 + 1)^3, \text{ then } dy = 3(x^2 + 1)^2 \cdot 2x \, dx = 6x(x^2 + 1)^2 \, dx \]

\[ \text{Differentials "chain together" naturally} \]

⚠️

Error Analysis and Accuracy

Understanding the accuracy of differential approximations:

\[ \text{Error} = |\Delta y - dy| = |f(x + dx) - f(x) - f'(x) \, dx| \]

\[ \text{Relative Error} = \frac{|\Delta y - dy|}{|\Delta y|} \times 100\% \]

\[ \text{Error} \approx \frac{1}{2}f''(x)(dx)^2 \quad \text{(Second-order approximation)} \]

\[ \text{Smaller dx values give more accurate approximations} \]

🌐

Multivariable Differentials

Differentials for functions of multiple variables:

\[ dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy \quad \text{(Total differential)} \]

\[ dz = f_x(x,y) \, dx + f_y(x,y) \, dy \]

\[ \text{For } z = f(x,y,w): \quad dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy + \frac{\partial z}{\partial w} dw \]

\[ \text{Each partial derivative shows sensitivity to one variable} \]

📏

Propagation of Uncertainty

Using differentials to estimate measurement errors:

\[ \text{If } z = f(x,y), \text{ then } |dz| \leq \left|\frac{\partial f}{\partial x}\right| |dx| + \left|\frac{\partial f}{\partial y}\right| |dy| \]

\[ \text{Maximum possible error bound} \]

\[ \text{Relative error: } \frac{|dz|}{|z|} \approx \left|\frac{\partial \ln f}{\partial x}\right| \frac{|dx|}{|x|} + \left|\frac{\partial \ln f}{\partial y}\right| \frac{|dy|}{|y|} \]

\[ \text{Used in experimental science and engineering tolerances} \]

🔢

Special Function Differentials

Differentials of other important functions:

\[ d(\sqrt{u}) = \frac{1}{2\sqrt{u}} \, du \]

\[ d\left(\frac{1}{u}\right) = -\frac{1}{u^2} \, du \]

\[ d(u^{1/n}) = \frac{1}{n} u^{(1/n)-1} \, du \]

\[ d(|u|) = \frac{u}{|u|} \, du \quad \text{(for } u \neq 0\text{)} \]

🎯 What does this mean?

Differentials are the mathematical tool for making "small change approximations" - they use the tangent line to estimate how much a function changes when inputs change slightly. Think of it as using a straight-line ruler to approximate a curved path over a short distance.

\[ dy \]

Differential of y - Approximate change in function value

\[ dx \]

Differential of x - Small change in input variable (same as Δx)

\[ f'(x) \]

Derivative - Rate of change multiplier for the differential

\[ \Delta y \]

Actual Change - True change in function value: f(x+Δx) - f(x)

\[ \Delta x \]

Change in x - Finite change in input variable

\[ L(x) \]

Linear Approximation - Tangent line function at point a

\[ a \]

Base Point - Point around which linearization is performed

\[ h \]

Small Increment - Small change from base point (h = x - a)

\[ \frac{\partial z}{\partial x} \]

Partial Derivative - Rate of change with respect to x, holding other variables constant

\[ dz \]

Total Differential - Approximate change in multivariable function

\[ u, v \]

Functions - Represent functions u(x) and v(x) in differential rules

\[ C \]

Constant - Any real number that doesn't change

\[ \sin u, \cos u \]

Trigonometric Functions - Functions of variable u in differential form

\[ e^u \]

Exponential Function - Natural exponential with variable u

\[ \ln u \]

Natural Logarithm - Logarithm of variable u

\[ \sqrt{u} \]

Square Root - Square root of variable u

🎯 Essential Insight: Differentials are the mathematical "magnifying glass" for small changes - they use the tangent line slope to predict how functions behave when inputs change slightly! 📊

🚀 Real-World Applications

🔬 Scientific Measurement & Error Analysis

Laboratory Precision & Uncertainty

Scientists use differentials to estimate how measurement errors in instruments propagate through calculations and affect final results

🏗️ Engineering & Manufacturing Tolerances

Quality Control & Design Specifications

Engineers use differentials to determine how small variations in part dimensions affect overall product performance and safety

💰 Economics & Finance

Sensitivity Analysis & Risk Assessment

Economists use differentials to estimate how small changes in interest rates, prices, or market conditions affect economic outcomes

🎯 Computer Graphics & Animation

Smooth Motion & Realistic Rendering

Game developers use differentials for smooth character movement, realistic physics simulations, and gradient-based lighting effects

The Magic: Science: Measurement errors → Uncertainty bounds, Engineering: Part tolerances → Product quality, Economics: Market changes → Impact prediction, Graphics: Small movements → Smooth animation

🎯

Master the "Small Change Estimator" Mindset!

Before diving into differential calculations, develop this core intuition:

Key Insight: Differentials are like having a mathematical "zoom lens" that lets you use straight-line approximations for curved functions over small distances - think of it as replacing a curved road with a straight ramp for short trips!

💡 Why this matters:

🔋 Real-World Power:

Science: Researchers estimate how measurement errors propagate through complex calculations
Engineering: Designers predict how small manufacturing variations affect product performance
Economics: Analysts estimate sensitivity of markets to small policy changes
Technology: Algorithms use differential approximations for optimization and machine learning

🧠 Mathematical Insight:

Differentials convert curved relationships into manageable linear approximations
Error analysis helps quantify the accuracy of approximations
Multivariable differentials handle complex systems with many inputs

🚀 Practice Strategy:

1 Understand the Approximation Concept 📐

Visualize: dy ≈ Δy for small changes in x
Think: "Tangent line slope × small change = approximate function change"
Key Formula: dy = f'(x) dx

2 Apply Linear Approximation 📊

Formula: f(x + dx) ≈ f(x) + f'(x) dx
Use known point values to estimate nearby values
Example: √9.1 ≈ √9 + (1/6)(0.1) = 3.0167

3 Master Differential Rules 🔗

Sum: d(u + v) = du + dv
Product: d(uv) = u dv + v du
Chain rule: dy = (dy/du)(du/dx) dx

4 Handle Error Analysis 📏

Calculate: |Error| = |Δy - dy|
Understand: Smaller dx gives better approximations
Apply: Use for uncertainty propagation in measurements

When you see differentials as the mathematical "small change calculator" that transforms curved problems into straight-line solutions, calculus becomes a practical tool for handling uncertainty and making accurate approximations in real-world scenarios!

Memory Trick: "Differentials Investigate Function Fluctuations Explaining Reality Estimating Numbers Through Incremental Approximation Logic Systems" - LINEAR: Straight-line approximation, SMALL: Works best for tiny changes, TANGENT: Uses slope of tangent line

🔑 Key Properties of Differentials

📐

Linear Approximation

dy = f'(x) dx provides best linear estimate of function change

Uses tangent line slope to approximate curved function behavior

🎯

Small Change Accuracy

Approximation accuracy improves as dx approaches zero

Error ≈ ½f''(x)(dx)² - quadratic in the increment size

🔗

Algebraic Properties

d(u + v) = du + dv; d(uv) = u dv + v du

Differentials follow same rules as derivatives

📏

Error Propagation

Shows how input uncertainties affect output uncertainties

Essential for experimental science and engineering tolerances

Universal Insight: Differentials are the bridge between theoretical calculus and practical problem-solving - they make curved relationships manageable using straight-line thinking!

Basic Formula: dy = f'(x) dx for linear approximation of function changes

Approximation Rule: f(x + dx) ≈ f(x) + f'(x) dx for small dx values

Error Analysis: Smaller changes give more accurate differential approximations

Multivariable Form: dz = (∂z/∂x) dx + (∂z/∂y) dy for functions of multiple variables

Trigonometric Differentials: d(sin u) = cos u du, d(cos u) = -sin u du

Exponential Differentials: d(e^u) = e^u du, d(ln u) = (1/u) du

Power Rule: d(u^n) = nu^(n-1) du for any real number n

Chain Rule Application: Differentials naturally follow the chain rule pattern

Integration by Partial Functions – Decomposition Method