Master all derivative rules with interactive examples, step-by-step solutions, and 50+ practice problems. This is your complete learning companion to differentiation rules.
This page is your interactive learning hub for derivative rules. Here's how it complements our other resources:
Quick reference sheet for all derivative formulas
Printable summary of essential derivative rules
Interactive learning with examples & practice problems
Advanced tool for complex calculations with graphs
This comprehensive table includes all essential derivative rules you'll need for calculus. Bookmark this page for quick reference when solving problems. Each rule includes the formula, example, and when to use it.
| Rule Name | Formula | Example | When to Use |
|---|---|---|---|
| Constant Rule | d/dx [c] = 0 |
d/dx [5] = 0 |
Differentiating any constant number |
| Power Rule | d/dx [xⁿ] = n·xⁿ⁻¹ |
d/dx [x³] = 3x² |
Any term with variable raised to power |
| Constant Multiple Rule | d/dx [c·f(x)] = c·f'(x) |
d/dx [3x²] = 3·2x = 6x |
Constant multiplied by function |
| Sum Rule | d/dx [f(x) + g(x)] = f'(x) + g'(x) |
d/dx [x² + x³] = 2x + 3x² |
Sum of two or more functions |
| Difference Rule | d/dx [f(x) - g(x)] = f'(x) - g'(x) |
d/dx [x³ - x²] = 3x² - 2x |
Difference between functions |
| Product Rule | d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x) |
d/dx [x·sin(x)] = 1·sin(x) + x·cos(x) |
Product of two functions |
| Quotient Rule | d/dx [f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]² |
d/dx [x²/(x+1)] = [2x·(x+1) - x²·1]/(x+1)² |
Division of two functions |
| Chain Rule | d/dx [f(g(x))] = f'(g(x))·g'(x) |
d/dx [sin(x²)] = cos(x²)·2x |
Composite functions (function of function) |
| Exponential Rule (eˣ) | d/dx [eˣ] = eˣ |
d/dx [e³ˣ] = 3e³ˣ |
Natural exponential functions |
| Logarithmic Rule (ln x) | d/dx [ln(x)] = 1/x |
d/dx [ln(x²)] = 2x/x² = 2/x |
Natural logarithm functions |
| Sine Rule | d/dx [sin(x)] = cos(x) |
d/dx [sin(2x)] = 2cos(2x) |
Sine trigonometric functions |
| Cosine Rule | d/dx [cos(x)] = -sin(x) |
d/dx [cos(x²)] = -2x·sin(x²) |
Cosine trigonometric functions |
| Tangent Rule | d/dx [tan(x)] = sec²(x) |
d/dx [tan(3x)] = 3sec²(3x) |
Tangent trigonometric functions |
| General Power Rule | d/dx [uⁿ] = n·uⁿ⁻¹·u' |
d/dx [(x²+1)³] = 3(x²+1)²·2x |
Power of a function (Chain + Power) |
| Implicit Differentiation | d/dx [f(y)] = f'(y)·dy/dx |
d/dx [y²] = 2y·dy/dx |
Equations not solved for y |
Pro Tip: Most real-world problems require combining multiple rules. For example, differentiating e^(x²)·sin(3x) requires Product Rule, Chain Rule, and Exponential Rule.
The Power Rule is the most fundamental derivative rule. It states that for any real number exponent n, the derivative of xⁿ is n·xⁿ⁻¹.
Solution: f'(x) = 4x³
Apply the rule directly: bring down exponent, subtract 1.
Solution: f'(x) = ½x^(-½) = 1/(2√x)
Rewrite radicals as fractional powers first.
Solution: f'(x) = -3x⁻⁴ = -3/x⁴
Same rule works for negative exponents.
The Power Rule can be derived using the limit definition of the derivative:
For f(x) = xⁿ, this becomes:
Substitute into the limit:
As h→0, all terms with h disappear:
The Product Rule is used when differentiating the product of two functions. The formula is often remembered as: "First times derivative of second, plus second times derivative of first."
Solution: f'(x) = 2x·sin(x) + x²·cos(x)
Solution: f'(x) = eˣ·ln(x) + eˣ·(1/x)
Solution: f'(x) = 1·(x²+3) + (x+1)·2x
For h(x) = x³·cos(x):
Sometimes it's easier to multiply first, then differentiate:
Then use Power Rule: f'(x) = 2x - 1 (easier than Product Rule).
The Quotient Rule is used for differentiating ratios of functions. A useful mnemonic is: "Low d-high minus high d-low, over low squared."
Solution: f'(x) = [2x·(x-1) - (x²+1)·1]/(x-1)²
Solution: f'(x) = [cos(x)·x - sin(x)·1]/x²
Solution: f'(x) = [eˣ·(x²+1) - eˣ·2x]/(x²+1)²
The Quotient Rule can be derived by rewriting the quotient as a product and using the Chain Rule:
When deciding which to use:
The Chain Rule handles composite functions (functions within functions). It's arguably the most important derivative rule because composite functions appear everywhere in advanced mathematics, physics, and engineering.
Solution: f'(x) = cos(3x²)·6x
Solution: f'(x) = e^(x³)·3x²
Solution: f'(x) = 5(2x+1)⁴·2
For triple compositions, apply Chain Rule multiple times:
This is: sin(□) where □ = e^(○) and ○ = x²
Most real problems combine Chain Rule with Product or Quotient Rule:
This needs Product Rule (for x²·sin(3x)) and Chain Rule (for sin(3x)).
All six trigonometric functions have specific derivative formulas. These are essential for physics, engineering, and any application involving periodic functions.
| Function | Derivative | Proof Method | Common Use |
|---|---|---|---|
| sin(x) | cos(x) |
Limit definition | Wave motion, oscillations |
| cos(x) | -sin(x) |
Limit definition | Phase shifts, AC circuits |
| tan(x) | sec²(x) |
Quotient Rule (sin/cos) | Slopes, angles |
| cot(x) | -csc²(x) |
Quotient Rule (cos/sin) | Complementary angles |
| sec(x) | sec(x)tan(x) |
Chain Rule (1/cos) | Trigonometric identities |
| csc(x) | -csc(x)cot(x) |
Chain Rule (1/sin) | Trigonometric identities |
Exponential and logarithmic functions have unique properties that make their derivatives particularly elegant and important for modeling growth, decay, and many natural phenomena.
The function f(x) = eˣ is remarkable because it is its own derivative: d/dx[eˣ] = eˣ.
Solution: f'(x) = e^(2x)·2 = 2e^(2x)
Chain Rule: derivative of exponent (2x) is 2
Solution: f'(x) = 5ˣ·ln(5)
Rewrite: 5ˣ = e^(x·ln5), then Chain Rule
Solution: f'(x) = eˣ + x·eˣ = eˣ(1+x)
Product Rule: (x)'·eˣ + x·(eˣ)'
For complex products/quotients/powers, logarithmic differentiation simplifies the process:
Direct differentiation is difficult.
When a function is not explicitly solved for y (e.g., x² + y² = 25), we use implicit differentiation. We differentiate both sides with respect to x, treating y as a function of x.
Solution: 2x + 2y·dy/dx = 0 ⇒ dy/dx = -x/y
Solution: 3x² + 3y²·dy/dx = 6y + 6x·dy/dx
Solution: cos(xy)·(y + x·dy/dx) = 1
Note: d/dx[y³] = 3y²·dy/dx (Chain Rule)
Second, third, and higher derivatives have important physical interpretations: velocity (1st derivative), acceleration (2nd), jerk (3rd), and beyond.
f'(x): 4x³
f''(x): 12x²
f'(x): cos(x)
f''(x): -sin(x)
f'''(x): -cos(x)
f'(x): 2e^(2x)
f''(x): 4e^(2x)
Pattern: f⁽ⁿ⁾(x) = 2ⁿ·e^(2x)
Concavity Test: f''(x) > 0 ⇒ concave up; f''(x) < 0 ⇒ concave down
Second Derivative Test for Extrema: At critical point c, if f''(c) > 0 ⇒ local minimum; f''(c) < 0 ⇒ local maximum
Test your understanding with these carefully selected practice problems. Start with basics, then progress to challenging combinations.
| # | Function | Required Rules | Solution (Click to Reveal) |
|---|---|---|---|
| 1 | f(x) = 3x⁵ - 2x³ + 7 |
Power, Constant Multiple, Sum | |
| 2 | f(x) = (2x+1)(x²-3) |
Product Rule or Expand First | |
| 3 | f(x) = (x²+1)/(x-2) |
Quotient Rule | |
| 4 | f(x) = √(3x²+1) |
Chain Rule, Power Rule | |
| 5 | f(x) = sin(2x)cos(3x) |
Product Rule, Chain Rule | |
| 6 | f(x) = e^(x²)·ln(x) |
Product Rule, Chain Rule, Exponential, Log | |
| 7 | f(x) = tan(√x) |
Chain Rule (twice), Tangent Rule | |
| 8 | f(x) = (2x³-1)⁴ |
Chain Rule, Power Rule | |
| 9 | f(x) = x·e^(-x²) |
Product Rule, Chain Rule, Exponential | |
| 10 | f(x) = ln(x²+1)/x |
Quotient Rule, Chain Rule, Log Rule |
For advanced students:
x³ + y³ = 3xy (Folium of Descartes)f(x) = x^(x^x) using logarithmic differentiationf(x) = sin(2x) + e^(3x)f(x) = ∫₀ˣ sin(t²) dt (Fundamental Theorem of Calculus)sin(x+y) = x² + y²Practice applying the rules you've learned with this interactive tool. Enter any function and see step-by-step solutions.
Quick examples: Click any to try
After finding a derivative, verify it using:
Our main calculator is designed for advanced calculations with features like 3D graphs, higher-order derivatives, and complex function analysis. This page is an interactive learning hub focused on teaching derivative rules with examples, practice problems, and step-by-step explanations. Think of it as your classroom, while the main calculator is your advanced toolkit.
Follow this decision tree:
Most problems require multiple rules applied in sequence.
This unique property comes from the definition of e as the base where the derivative of aˣ equals itself. Mathematically:
The limit lim_(h→0) [eʰ - 1]/h = 1 is actually the definition of the number e. This self-replicating property makes exponential functions essential for modeling continuous growth and decay.
Follow this 4-step mastery plan:
Consistency is key - 30 minutes daily is better than 5 hours once a week.
Explore our other calculus tools to continue your learning journey: