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.

These rules fall 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: $\frac{d}{dx}(c) = 0$ 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 $\sqrt{7}$. These are all just numbers, so 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 $x^{-3}$, then apply the rule normally
• This is the 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 $f'(x) = g'(x) \pm h'(x)$
• Constant multipliers pull out freely: $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$. This property, combined with term-by-term differentiation, is called the linearity of the derivative
• 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 factor intact
• The order of the two terms doesn't affect the final answer, but picking a consistent order prevents errors
• Common exam setup: functions given as tables or graphs where you must apply Product Rule at a specific $x$-value. You won't have a formula, just the values of $u$, $v$, $u'$, and $v'$ at that point

Quotient Rule

• Formula: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$. A common mnemonic is "low d-high minus high d-low, over low squared"
• Numerator order matters. Switching $u'v$ and $uv'$ flips the sign of the entire numerator, giving you the wrong answer
• Alternative approach: rewrite $\frac{u}{v}$ as $u \cdot v^{-1}$ and use Product Rule with Chain Rule instead. Many students find this reduces sign errors

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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 function is $\sin(\square)$ and the inside is $x^2$. You differentiate $\sin$ to get $\cos(x^2)$, then multiply by $\frac{d}{dx}(x^2) = 2x$, giving $2x\cos(x^2)$
• Most commonly missed rule. Forgetting to multiply by $g'(x)$ is the single most frequent 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

• $\frac{d}{dx}(e^x) = e^x$. This is 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 constant multiplier
• With Chain Rule: $\frac{d}{dx}(e^{g(x)}) = e^{g(x)} \cdot g'(x)$. For example, $\frac{d}{dx}(e^{3x}) = e^{3x} \cdot 3 = 3e^{3x}$

Natural Logarithm Rule

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$ for $x > 0$. One of the cleanest derivative formulas
• For other bases: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$. You can also convert to natural log first using change of base, then differentiate
• With Chain Rule: $\frac{d}{dx}(\ln(g(x))) = \frac{g'(x)}{g(x)}$. This pattern appears constantly in implicit differentiation and logarithmic 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 sign appears with cosine's derivative, not sine's
• $\frac{d}{dx}(\tan x) = \sec^2 x$. You can verify this yourself by applying Quotient Rule to $\frac{\sin x}{\cos x}$
• Cyclic pattern: differentiating $\sin x$ four times in a row gives $\cos x \to -\sin x \to -\cos x \to \sin x$, cycling back to the start

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 here. This form also shows up frequently in integration
• Domain restrictions matter. These derivatives are only valid where the original inverse functions are defined (for example, $\arcsin x$ requires $-1 < x < 1$)

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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 $3^x$. How does the constant 3 appear differently 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?