Important Differentiation Rules

Why This Matters

Differentiation rules aren't just formulas to memorize—they're the fundamental toolkit that lets you analyze how any function changes. In mathematical analysis, you're being tested on your ability to recognize which rule applies, execute it correctly, and combine multiple rules when functions get complex. The real exam challenge isn't remembering that the power rule exists; it's knowing when you need the chain rule nested inside a product rule, or recognizing that logarithmic differentiation will save you twenty minutes of algebra.

These rules demonstrate core principles of calculus: linearity of the derivative, how composition affects rates of change, and the relationship between algebraic structure and differentiation strategy. When you see a function on an exam, your first job is pattern recognition—what's the structure? Is it a product, a composition, an implicit relationship? Don't just memorize formulas—know what structural feature of a function triggers each rule.

---

Basic Building Blocks

These foundational rules handle the simplest function types and form the basis for everything else. The derivative operator is linear, meaning it respects addition and scalar multiplication—this is why these rules work so cleanly.

Constant Rule

• The derivative of any constant is zero—constants don't change, so their rate of change is nothing
• Formal statement: if $f(x) = c$ for any constant $c$, then $f'(x) = 0$
• Conceptual foundation: this reflects that derivatives measure change, and constants are unchanging by definition

Power Rule

• Core formula: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$—bring down the exponent, reduce by one
• Universal application: works for any real exponent $n$, including negatives and fractions
• Most common exam errors: forgetting to apply this when $n = 1$ (derivative is just 1) or $n = 0$ (derivative is 0)

Sum and Difference Rule

• Linearity in action: the derivative of $f(x) + g(x)$ equals $f'(x) + g'(x)$
• Extends to differences: subtraction works identically—$\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$
• Strategic value: lets you break apart complex expressions and differentiate term by term

---

Combining Functions: Products and Quotients

When functions are multiplied or divided, simple term-by-term differentiation fails. These rules account for how both components contribute to the overall rate of change.

Product Rule

• The formula: if $f(x) = g(x) \cdot h(x)$, then $f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$
• Memory device: "derivative of first times second, plus first times derivative of second"
• Why it works: both factors are changing simultaneously, so you must account for each contribution separately

Quotient Rule

• The formula: if $f(x) = \frac{g(x)}{h(x)}$, then $f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{[h(x)]^2}$
• Memory device: "low d-high minus high d-low, over low squared" (where "high" is numerator, "low" is denominator)
• Alternative approach: rewrite as $g(x) \cdot [h(x)]^{-1}$ and use product rule with chain rule instead

---

Handling Composition: The Chain Rule

Composite functions—functions inside other functions—require tracking how changes propagate through each layer. The chain rule is arguably the most important rule because it appears constantly in combination with others.

Chain Rule

• The formula: if $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$—derivative of outside times derivative of inside
• Recognition key: ask yourself "is there a function inside another function?" If yes, you need chain rule
• Nested applications: for deeply composed functions like $\sin(e^{x^2})$, apply chain rule multiple times, working outside-in

Exponential Function Rule

• Base $e$ case: $\frac{d}{dx}[e^{f(x)}] = e^{f(x)} \cdot f'(x)$—the exponential survives, multiplied by inner derivative
• General base: $\frac{d}{dx}[a^{f(x)}] = a^{f(x)} \cdot \ln(a) \cdot f'(x)$—the $\ln(a)$ factor appears for non-$e$ bases
• Why $e$ is special: it's the unique base where the derivative of $e^x$ is simply $e^x$—no extra constant needed

Trigonometric Function Rules

• Core derivatives: $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$, $\frac{d}{dx}(\tan x) = \sec^2 x$
• Pattern recognition: sine and cosine cycle into each other (with sign changes); tangent becomes secant squared
• Chain rule integration: for $\sin(f(x))$, the derivative is $\cos(f(x)) \cdot f'(x)$—always multiply by inner derivative

---

Advanced Techniques

When standard rules become unwieldy, these techniques offer strategic alternatives. They're not separate rules but rather clever applications of existing rules to handle difficult cases.

Implicit Differentiation

• When to use: when $y$ is defined implicitly by an equation like $x^2 + y^2 = 1$ rather than explicitly as $y = f(x)$
• Method: differentiate both sides with respect to $x$, treating $y$ as a function of $x$ (so $\frac{d}{dx}[y^2] = 2y \cdot \frac{dy}{dx}$)
• Solve for $\frac{dy}{dx}$: after differentiating, algebraically isolate $\frac{dy}{dx}$ to get your answer

Logarithmic Differentiation

• Strategic advantage: transforms products into sums and powers into coefficients via logarithm properties
• Method: take $\ln$ of both sides, differentiate implicitly, then solve for $\frac{dy}{dx}$ and substitute back
• Best applications: functions like $y = x^x$ or $y = \frac{(x+1)^3(x-2)^4}{(x+5)^2}$ where direct rules are messy

---

Quick Reference Table

---

Self-Check Questions

1. Which two rules both handle situations where two functions are combined algebraically, and what's the key structural difference between when you use each?

2. You encounter $f(x) = e^{\sin(x^2)}$. Which rules do you need, and in what order do you apply them?

3. Compare and contrast implicit differentiation and logarithmic differentiation: when is each technique the better choice, and what do they have in common procedurally?

4. Why does the quotient rule have a minus sign while the product rule has a plus sign? What would go wrong if you used a plus sign in the quotient rule?

5. If an FRQ gives you $y = x^{\cos x}$ and asks for $\frac{dy}{dx}$, which technique should you reach for first, and why do standard rules fail here?