Common Derivatives Formulas

Why This Matters

Derivative formulas are the building blocks of calculus—they're what you reach for every single time you need to find a rate of change, analyze a function's behavior, or solve an optimization problem. You're being tested not just on whether you can recall these formulas, but on whether you can recognize which rule applies and combine multiple rules in a single problem. The chain rule nested inside a product rule? That's where exams separate students who memorized from those who understood.

These formulas connect directly to bigger concepts: instantaneous rates of change, slope functions, optimization, and modeling real-world phenomena. When you see a growth model in statistics or a velocity function in physics, you're applying these exact rules. Don't just memorize the formulas—know when each one triggers and how they work together. That's what turns a tricky FRQ into points on the board.

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Basic Building Blocks

These foundational rules handle the simplest cases and form the basis for everything else. Master these first, because every complex derivative breaks down into these components.

Power Rule

• $\frac{d}{dx}(x^n) = nx^{n-1}$—works for any real exponent, including negatives and fractions
• Polynomial differentiation becomes trivial: bring down the exponent, reduce by one
• Foundation for integration reversal—recognizing power rule patterns helps you antidifferentiate quickly

Constant Rule

• $\frac{d}{dx}(c) = 0$—constants have no rate of change, so their derivative vanishes
• Simplifies complex expressions by eliminating constant terms immediately
• Conceptual meaning: a horizontal line has zero slope everywhere

Sum and Difference Rule

• $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$—differentiate each term independently
• Linearity property allows you to break apart complicated expressions
• Pairs with constant multiple rule: $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$ for any constant $c$

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Combination Rules

When functions are multiplied, divided, or composed, these rules tell you how to handle the interaction. The key insight: derivatives don't "distribute" over products and quotients the way you might hope.

Product Rule

• $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$—"derivative of first times second, plus first times derivative of second"
• Common mnemonic: if $u$ and $v$ are functions, then $(uv)' = u'v + uv'$
• Watch for: problems where one factor is simpler (like $x$) make computation easier when you differentiate that one first

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"
• Numerator order matters: subtraction isn't commutative, so getting this backwards flips your sign
• Alternative approach: rewrite as $f(x) \cdot [g(x)]^{-1}$ and use product rule with chain rule instead

Chain Rule

• $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$—differentiate outside, keep inside, then multiply by derivative of inside
• Most commonly tested rule because it appears in nearly every non-trivial derivative problem
• Nested applications: for $f(g(h(x)))$, work from outside in, multiplying derivatives at each layer

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Exponential and Logarithmic Functions

These derivatives appear constantly in growth/decay models and statistical distributions. The natural base $e$ has special properties that make these formulas elegant.

Natural Exponential Function

• $\frac{d}{dx}(e^x) = e^x$—the only function that is its own derivative
• General exponential: $\frac{d}{dx}(a^x) = a^x \ln(a)$ for any positive base $a$
• Growth models in statistics (population, compound interest) rely on this self-replicating property

Natural Logarithm

• $\frac{d}{dx}(\ln x) = \frac{1}{x}$—valid only for $x > 0$
• General logarithm: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$
• Inverse relationship: since $\ln x$ and $e^x$ are inverses, their derivatives are reciprocally related

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Trigonometric Functions

Trig derivatives follow cyclical patterns. Notice how sine and cosine derivatives cycle through each other, and how the "co-" functions always pick up a negative sign.

Sine and Cosine

• $\frac{d}{dx}(\sin x) = \cos x$ and $\frac{d}{dx}(\cos x) = -\sin x$—the negative appears on cosine's derivative
• Cyclical pattern: differentiating four times returns you to the original function
• Unit circle connection: derivative represents how fast the $y$-coordinate (sine) or $x$-coordinate (cosine) changes

Tangent, Secant, and Others

• $\frac{d}{dx}(\tan x) = \sec^2 x$—can be derived using quotient rule on $\frac{\sin x}{\cos x}$
• $\frac{d}{dx}(\sec x) = \sec x \tan x$ and $\frac{d}{dx}(\csc x) = -\csc x \cot x$
• Pattern recognition: "co-" functions (cosine, cotangent, cosecant) always have negative derivatives

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Inverse Trigonometric Functions

These derivatives look intimidating but follow predictable patterns. They appear frequently in integration problems and anywhere you're "undoing" a trig function.

Arcsine and Arccosine

• $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$—defined for $-1 < x < 1$
• $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$—same magnitude, opposite sign
• Domain restriction matters: these derivatives blow up as $x \to \pm 1$

Arctangent

• $\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$—defined for all real $x$
• No square root in denominator makes this one easier to work with
• Integration connection: recognizing $\frac{1}{1+x^2}$ as the derivative of $\arctan x$ is a common integration technique

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Quick Reference Table

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Self-Check Questions

1. Which two rules would you combine to differentiate $f(x) = e^{x^2}$, and in what order do you apply them?

2. The derivatives of $\arcsin x$ and $\arccos x$ have the same denominator but opposite signs. Why does this make sense given the relationship between these two functions?

3. Compare and contrast the product rule and chain rule: give an example of a function requiring each, and explain how you identify which rule applies.

4. If $\frac{d}{dx}(\sin x) = \cos x$, why does $\frac{d}{dx}(\cos x)$ have a negative sign? (Hint: think about the graphs.)

5. An FRQ asks you to find the derivative of $f(x) = \frac{x^2 \ln x}{e^x}$. List all the derivative rules you'll need and describe your strategy for combining them.