Common Derivative Rules

Why This Matters

Derivative rules are the fundamental toolkit you'll use throughout calculus—they're not just formulas to memorize, but logical shortcuts that let you analyze how any function changes. You're being tested on your ability to recognize which rule applies, when to combine multiple rules, and how to execute them accurately under pressure. Every optimization problem, related rates question, and curve-sketching exercise depends on your fluency with these rules.

Think of these rules as falling into categories: some handle basic building blocks (constants, powers), others manage function combinations (sums, products, quotients), and the rest tackle special function families (exponential, logarithmic, trigonometric). Don't just memorize the formulas—know what type of function structure triggers each rule, and practice recognizing when you need to chain multiple rules together. That's where exam points are won or lost.

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Basic Building Block Rules

These rules handle the simplest function types and form the foundation for everything else. Master these first, because you'll use them constantly inside more complex problems.

Constant Rule

• The derivative of any constant is zero—written as $\frac{d}{dx}(c) = 0$ for any constant $c$
• Constants don't change, so their rate of change is zero by definition
• Watch for hidden constants like $\pi$, $e^2$, or $\sqrt{7}$—they all differentiate to zero

Power Rule

• The derivative of $x^n$ is $nx^{n-1}$—this works for any real exponent, including negatives and fractions
• Rewrite before differentiating: express $\sqrt{x}$ as $x^{1/2}$ and $\frac{1}{x^3}$ as $x^{-3}$
• Most-tested rule in calculus—you'll apply it dozens of times per exam, often combined with Chain Rule

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Combination Rules

When functions are added, multiplied, or divided, these rules tell you how to handle the structure. The key is recognizing the operation connecting the parts.

Sum and Difference Rule

• Differentiate term by term—if $f(x) = g(x) \pm h(x)$, then $f'(x) = g'(x) \pm h'(x)$
• Linearity of derivatives means you can also pull out constant multipliers: $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$
• Break complex expressions apart before applying other rules to each piece

Product Rule

• Formula: $(uv)' = u'v + uv'$—differentiate each factor once while keeping the other intact
• Order doesn't matter for the final answer, but consistency prevents errors
• Common FRQ setup: functions given as tables or graphs where you must apply Product Rule at a specific point

Quotient Rule

• Formula: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$—remember "low d-high minus high d-low, over low squared"
• Numerator order matters—switching $u'v$ and $uv'$ gives the wrong sign
• Alternative approach: rewrite $\frac{u}{v}$ as $u \cdot v^{-1}$ and use Product Rule with Chain Rule instead

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The Chain Rule

This is the rule that connects everything else. Whenever one function is "inside" another, Chain Rule is required.

Chain Rule

• Formula: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$—differentiate the outer function, keep the inner function unchanged, then multiply by the inner function's derivative
• "Outside-inside" thinking: for $\sin(x^2)$, the outside is $\sin(\square)$ and inside is $x^2$
• Most commonly missed rule—forgetting to multiply by $g'(x)$ is the #1 derivative error on exams

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Exponential and Logarithmic Rules

These rules handle functions where the variable appears in an exponent or inside a logarithm. The natural base $e$ has special properties that simplify calculations.

Exponential Function Rule

• $\frac{d}{dx}(e^x) = e^x$—the only function that equals its own derivative
• For other bases: $\frac{d}{dx}(a^x) = a^x \ln(a)$—the natural log of the base appears as a multiplier
• With Chain Rule: $\frac{d}{dx}(e^{g(x)}) = e^{g(x)} \cdot g'(x)$—don't forget the inner derivative

Natural Logarithm Rule

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$ for $x > 0$—one of the cleanest derivative formulas
• For other bases: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$—convert using change of base if needed
• With Chain Rule: $\frac{d}{dx}(\ln(g(x))) = \frac{g'(x)}{g(x)}$—this pattern appears constantly in implicit differentiation

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Trigonometric Function Rules

These rules govern periodic functions essential for modeling waves, oscillations, and circular motion. Notice the sign patterns and how cofunctions relate.

Basic Trig Derivatives (sin, cos, tan)

• $\frac{d}{dx}(\sin x) = \cos x$ and $\frac{d}{dx}(\cos x) = -\sin x$—the negative appears with cosine
• $\frac{d}{dx}(\tan x) = \sec^2 x$—derived from Quotient Rule on $\frac{\sin x}{\cos x}$
• Pattern recognition: derivatives cycle through trig functions; taking four derivatives of $\sin x$ returns to $\sin x$

Inverse Trig Derivatives

• $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$ and $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$—same magnitude, opposite signs
• $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$—no square root, and this form appears frequently in integration
• Domain restrictions matter—these derivatives are only valid where the original inverse functions are defined

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Quick Reference Table

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Self-Check Questions

1. Which two rules would you combine to differentiate $f(x) = (2x+1)^4 \cdot e^x$?

2. Compare the derivatives of $e^{3x}$ and $3^x$—what's different about how the constant 3 appears in each result?

3. A student differentiates $\ln(x^2+1)$ and gets $\frac{1}{x^2+1}$. What did they forget, and what's the correct answer?

4. If an FRQ gives you $f(x) = \frac{x^2 \sin x}{e^x}$, which rules do you need, and in what order would you apply them?

5. Compare $\frac{d}{dx}(\sin^2 x)$ and $\frac{d}{dx}(\sin(x^2))$—both require Chain Rule, but what's the "inner function" in each case?