Complete Derivative Rules Guide
Master all essential derivative rules with clear formulas, examples, and explanations. Your complete reference for calculus success.
📐 Basic Derivative Rules
1. Power Rule
The most fundamental and frequently used rule in calculus.
d/dx[xⁿ] = n·xⁿ⁻¹
Examples:
d/dx[x³] = 3x²
d/dx[x⁵] = 5x⁴
d/dx[x] = 1
d/dx[√x] = 1/(2√x)
d/dx[x³] = 3x²
d/dx[x⁵] = 5x⁴
d/dx[x] = 1
d/dx[√x] = 1/(2√x)
2. Constant Rule
The derivative of any constant is always zero.
d/dx[c] = 0
Examples:
d/dx[7] = 0
d/dx[-15] = 0
d/dx[π] = 0
d/dx[7] = 0
d/dx[-15] = 0
d/dx[π] = 0
3. Constant Multiple Rule
Constants can be pulled out of derivatives.
d/dx[c·f(x)] = c·f'(x)
Examples:
d/dx[5x³] = 5·3x² = 15x²
d/dx[-2x⁴] = -2·4x³ = -8x³
d/dx[πx²] = π·2x = 2πx
d/dx[5x³] = 5·3x² = 15x²
d/dx[-2x⁴] = -2·4x³ = -8x³
d/dx[πx²] = π·2x = 2πx
4. Sum/Difference Rule
Take derivatives term by term.
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
Examples:
d/dx[x³ + x²] = 3x² + 2x
d/dx[5x⁴ - 3x²] = 20x³ - 6x
d/dx[x⁵ + 2x³ - 7x] = 5x⁴ + 6x² - 7
d/dx[x³ + x²] = 3x² + 2x
d/dx[5x⁴ - 3x²] = 20x³ - 6x
d/dx[x⁵ + 2x³ - 7x] = 5x⁴ + 6x² - 7
🔗 Advanced Rules
5. Product Rule
For products of two functions.
d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Example:
d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x)
d/dx[x²·sin(x)] = 2x·sin(x) + x²·cos(x)
6. Quotient Rule
For quotients of two functions.
d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²
Example:
d/dx[x²/sin(x)] = [2x·sin(x) - x²·cos(x)] / sin²(x)
d/dx[x²/sin(x)] = [2x·sin(x) - x²·cos(x)] / sin²(x)
7. Chain Rule
For composite functions - most important rule!
d/dx[f(g(x))] = f'(g(x))·g'(x)
Examples:
d/dx[sin(x²)] = cos(x²)·2x
d/dx[(x² + 1)⁵] = 5(x² + 1)⁴·2x
d/dx[e^(3x)] = e^(3x)·3
d/dx[sin(x²)] = cos(x²)·2x
d/dx[(x² + 1)⁵] = 5(x² + 1)⁴·2x
d/dx[e^(3x)] = e^(3x)·3
📈 Special Functions
Exponential Functions
d/dx[eˣ] = eˣ
d/dx[aˣ] = aˣ·ln(a)
d/dx[aˣ] = aˣ·ln(a)
Logarithmic Functions
d/dx[ln(x)] = 1/x
d/dx[log_a(x)] = 1/(x·ln(a))
d/dx[log_a(x)] = 1/(x·ln(a))
Trigonometric Functions
d/dx[sin(x)] = cos(x)
d/dx[cos(x)] = -sin(x)
d/dx[tan(x)] = sec²(x)
d/dx[cot(x)] = -csc²(x)
d/dx[sec(x)] = sec(x)tan(x)
d/dx[csc(x)] = -csc(x)cot(x)
d/dx[cos(x)] = -sin(x)
d/dx[tan(x)] = sec²(x)
d/dx[cot(x)] = -csc²(x)
d/dx[sec(x)] = sec(x)tan(x)
d/dx[csc(x)] = -csc(x)cot(x)
💡 Key Tip: The chain rule is used in over 80% of derivative problems. Master it first, and the rest becomes easier!
⚠️ Common Mistake: Don't forget the chain rule when there's a function inside another function. d/dx[sin(2x)] = cos(2x)·2, NOT just cos(2x)!
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