James Zhang: Bilby live helps to teach calculus in an efficient and affective way.

"Calculus is the music of change, and these formulas are its most beautiful melodies."

Chapter 1: The Poetry of Derivatives

The Foundation: Power Rule

The power rule is where calculus begins to sing. When we have a function like $f(x) = x^3$, we're asking: "How fast is this changing at any point?"

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

Example: If $f(x) = x^3$, then $f'(x) = 3x^2$

Think of it this way: If you're climbing a mountain shaped like $x^3$, the power rule tells you exactly how steep the slope is at any point $x$. The higher you go, the steeper it gets!

Constants: The Steady Notes

Constants are the steady bass notes in our calculus symphony:

• $\frac{d}{dx}(c) = 0$ - Constants don't change, so their rate of change is zero

• $\frac{d}{dx}(cx) = c$ - Constants just tag along for the ride

The Chain Rule: Calculus's Most Elegant Dance

The chain rule is perhaps the most beautiful concept in calculus. It tells us how to handle functions within functions:

$$\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$$

Example: For $h(x) = \sin(x^2)$:

• Inner function: $g(x) = x^2$, so $g'(x) = 2x$

• Outer function: $f(u) = \sin(u)$, so $f'(u) = \cos(u)$

• Result: $h'(x) = \cos(x^2) \cdot 2x = 2x\cos(x^2)$

Imagine: You're watching a dancer spinning while moving in a circle. The chain rule captures both motions simultaneously!

Product and Quotient Rules: When Functions Dance Together

When two functions multiply, they create something beautiful:

$$\frac{d}{dx}(uv) = u'v + uv'$$

The Story: Each function takes turns being the "star" while the other provides support. First $u$ changes while $v$ stays constant, then $v$ changes while $u$ stays constant.

For division, we get the elegant quotient rule:

$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$$

The Transcendental Functions: Nature's Own Curves

These functions appear everywhere in nature:

• $\frac{d}{dx}(\sin x) = \cos x$ - The sine wave's slope is always the cosine

• $\frac{d}{dx}(\cos x) = -\sin x$ - A beautiful circular relationship

• $\frac{d}{dx}(e^x) = e^x$ - The only function that is its own derivative!

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$ - The natural logarithm's elegant simplicity

Chapter 2: The Art of Integration

Integration is the reverse of differentiation - like reconstructing a melody from its rhythm.

The Fundamental Integration Rules

$$\int x^n , dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

The Exception: $\int \frac{1}{x} , dx = \ln|x| + C$

Why the exception? Because $\frac{1}{x}$ is special - it's the derivative of the natural logarithm!

Trigonometric Integrals: Waves of Beauty

• $\int \sin x , dx = -\cos x + C$

• $\int \cos x , dx = \sin x + C$

A Beautiful Pattern: Notice how sine and cosine chase each other in an eternal dance through differentiation and integration.

Integration by Parts: The Elegant Exchange

$$\int u , dv = uv - \int v , du$$

The Mantra: "LIATE" - choose $u$ in this order:

• Logarithmic

• Inverse trigonometric

• Algebraic

• Trigonometric

• Exponential

Example: $\int x e^x , dx$

• Let $u = x$ (algebraic), $dv = e^x dx$

• Then $du = dx$, $v = e^x$

• Result: $xe^x - \int e^x dx = xe^x - e^x + C = e^x(x-1) + C$

Chapter 3: The Fundamental Theorem - Calculus's Crown Jewel

The Fundamental Theorem of Calculus connects derivatives and integrals in breathtaking harmony:

$$\int_a^b f(x) , dx = F(b) - F(a)$$

where $F'(x) = f(x)$

The Magic: This theorem tells us that the area under a curve from $a$ to $b$ equals the difference in antiderivative values. It's like saying the total change equals the final position minus the initial position!

Chapter 4: Limits - The Foundation of Everything

Limits are the philosophical foundation of calculus. They answer the question: "What happens as we approach, but never quite reach, a point?"

The Most Important Limits

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

This limit is the reason why $\frac{d}{dx}(\sin x) = \cos x$. It's the heartbeat of trigonometric calculus!

$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$$

This limit defines the most important number in calculus: Euler's number $e \approx 2.718$.

The Definition of a Derivative

$$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = f'(x)$$

This is poetry: We're asking what happens to the slope of a secant line as it becomes a tangent line.

Chapter 5: Series - Infinite Beauty

Power series let us express complex functions as infinite polynomials:

The Exponential Series

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$

The Trigonometric Series

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$

$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$

The Beauty: Notice how sine only has odd powers and cosine only has even powers - a perfect mathematical symmetry!

The Geometric Series

$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \quad (|x| < 1)$$

This series is the foundation for understanding convergence and divergence.

Practice Problems to Illuminate Understanding

Problem 1: The Chain Rule in Action

Find $\frac{d}{dx}[\ln(x^2 + 1)]$

Solution: Using the chain rule:

• Inner function: $u = x^2 + 1$, so $\frac{du}{dx} = 2x$

• Outer function: $\ln(u)$, so $\frac{d}{du}[\ln(u)] = \frac{1}{u}$

• Result: $\frac{1}{x^2 + 1} \cdot 2x = \frac{2x}{x^2 + 1}$

Problem 2: Integration by Parts

Find $\int x \cos x , dx$

Solution:

• Let $u = x$, $dv = \cos x , dx$

• Then $du = dx$, $v = \sin x$

• Result: $x \sin x - \int \sin x , dx = x \sin x + \cos x + C$

The Beauty of Calculus: A Final Reflection

Calculus is not just about computation - it's about understanding the language of change and motion. Every formula tells a story:

• Derivatives tell us about rates of change

• Integrals tell us about accumulation

• Limits tell us about approaching perfection

• Series tell us that infinity can be tamed

These formulas are not just tools; they're the vocabulary of the universe itself. From the orbit of planets to the flow of rivers, from the growth of populations to the decay of radioactive elements - calculus describes it all.

Master these formulas, and you hold the keys to understanding the mathematical poetry of our world.