Fundamental Calculus Formulas

Why This Matters

Calculus is the mathematical language of change, and in the physical sciences, everything changes—particles move, waves oscillate, populations grow, and energy transforms. You're being tested not just on whether you can apply these formulas mechanically, but on whether you understand when and why each rule applies. The differentiation rules tell you how quantities change instantaneously, while integration connects those rates back to accumulated totals.

These formulas fall into distinct categories: basic building blocks, combination rules, special functions, and the bridge between derivatives and integrals. Don't just memorize the formulas in isolation—know which rule handles which situation. When you see a product of functions, your brain should immediately flag the product rule. When you see a function inside another function, chain rule. That pattern recognition is what separates students who struggle on exams from those who breeze through them.

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Basic Building Blocks

These are the foundation of all differentiation. Master these first, because every other rule builds on them. The derivative measures instantaneous rate of change—these rules tell you the rate of change for the simplest possible functions.

Derivative of a Constant

• $\frac{d}{dx}(c) = 0$—constants don't change, so their rate of change is zero
• Graphical interpretation: a horizontal line has zero slope everywhere
• Physical meaning: if a quantity isn't varying with your independent variable, its derivative vanishes

Power Rule

• $\frac{d}{dx}(x^n) = nx^{n-1}$—works for any real exponent, including negatives and fractions
• Mechanism: bring down the exponent as a coefficient, then reduce the exponent by one
• Applications: polynomial functions, root functions ($x^{1/2}$), and inverse powers ($x^{-1}$)

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Combination Rules

When functions are combined through addition, multiplication, or division, these rules tell you how to break down the problem. The key insight: differentiation is linear for sums but requires special handling for products and quotients.

Sum Rule

• $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$—differentiate term by term
• Linearity property: this also works for differences and scalar multiples
• Exam strategy: always simplify expressions into sums before differentiating when possible

Product Rule

• $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$—"first times derivative of second, plus second times derivative of first"
• Memory trick: each function gets a turn being differentiated while the other stays unchanged
• Common errors: forgetting to include both terms—the derivative of a product is never just the product of derivatives

Quotient Rule

• $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$—"low d-high minus high d-low, over low squared"
• Sign matters: the numerator has subtraction, not addition (unlike the product rule)
• Alternative approach: rewrite as $f(x) \cdot [g(x)]^{-1}$ and use product rule with chain rule instead

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Composite Functions

When one function is nested inside another, you need the chain rule. This is arguably the most important rule for physical science applications, where quantities often depend on intermediate variables.

Chain Rule

• $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$—derivative of outer times derivative of inner
• Identification: look for "function of a function" structures like $\sin(x^2)$ or $e^{3x}$
• Physical interpretation: connects rates through intermediate variables (if y depends on u, and u depends on x, how does y depend on x?)

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Special Functions

These derivatives appear constantly in physical science applications. Exponential, logarithmic, and trigonometric functions model growth, decay, and oscillation—the three most common behaviors in nature.

Exponential Function

• $\frac{d}{dx}(e^x) = e^x$—the only function that equals its own derivative
• Why $e$ is special: this self-replicating property is why $e$ appears naturally in growth/decay problems
• With chain rule: $\frac{d}{dx}(e^{u}) = e^{u} \cdot \frac{du}{dx}$—don't forget the inner derivative

Natural Logarithm

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$—the antiderivative of $\frac{1}{x}$ is $\ln|x|$
• Inverse relationship: logarithms and exponentials are inverses, reflected in their derivative relationship
• Domain restriction: only valid for $x > 0$ (or use $\ln|x|$ for negative values)

Trigonometric Functions

• $\frac{d}{dx}(\sin x) = \cos x$ and $\frac{d}{dx}(\cos x) = -\sin x$—note the negative sign on cosine's derivative
• Cyclic pattern: differentiating four times returns you to the original function
• Physical applications: simple harmonic motion, wave equations, and anything oscillatory

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The Bridge: Connecting Derivatives and Integrals

This theorem is the conceptual heart of calculus—it tells you that differentiation and integration are inverse operations.

Fundamental Theorem of Calculus

• $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F'(x) = f(x)$—evaluate the antiderivative at the bounds and subtract
• Two-part theorem: Part 1 says integration and differentiation undo each other; Part 2 gives the evaluation formula
• Physical meaning: the integral accumulates the rate of change to recover the total change in a quantity

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Quick Reference Table

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Self-Check Questions

1. Which two rules both involve handling combinations of functions, but one uses addition in its formula while the other uses subtraction? What's the key structural difference between when you'd use each?

2. You need to differentiate $e^{x^2}$. Which rules do you need, and in what order do you apply them?

3. Compare and contrast: How are the derivatives of $e^x$ and $\sin x$ similar in their "self-referential" behavior? How do they differ?

4. If you're given a velocity function $v(t)$ and asked to find total displacement over an interval, which formula applies? What if you're given position $x(t)$ and asked for instantaneous velocity?

5. A classmate claims that $\frac{d}{dx}[\sin x \cdot \cos x] = \cos x \cdot (-\sin x)$. Identify their error and provide the correct derivative.