Differentiation Rules- A Complete Reference Guide
Differentiation Rules: The Complete Reference
Differentiation is the backbone of calculus. If you can't differentiate reliably, everything else in calculus falls apart. This guide cuts through the noise and gives you every rule you need, explained plainly.
What Differentiation Actually Is
Differentiation finds the instantaneous rate of change of a function. Visually, it gives you the slope of a curve at any point. Mathematically, if you have f(x), its derivative is f'(x) or df/dx.
You calculate derivatives using a set of rules. Memorize them. Use them. That's the game.
The Power Rule
The most fundamental rule. If f(x) = xⁿ, then:
f'(x) = nxⁿ⁻¹
That's it. Bring down the exponent, subtract one from it.
Examples
- f(x) = x⁵ → f'(x) = 5x⁴
- f(x) = x³ → f'(x) = 3x²
- f(x) = x² → f'(x) = 2x
- f(x) = x¹ → f'(x) = 1
- f(x) = x⁰ → f'(x) = 0
The power rule works for any real exponent, including negatives and fractions.
- f(x) = x⁻² → f'(x) = -2x⁻³
- f(x) = x^(1/2) → f'(x) = (1/2)x^(-1/2)
Constant Rules
Constant multiple: If f(x) = c·g(x), then f'(x) = c·g'(x). Pull out the constant.
Example: f(x) = 7x³ → f'(x) = 7·3x² = 21x²
Constant rule: The derivative of any constant is zero. f(x) = 5 → f'(x) = 0. A flat line has no slope.
Sum and Difference Rules
The derivative of a sum is the sum of the derivatives. Same for differences.
f'(x) ± g'(x)
Example: f(x) = 3x² + 5x - 7
f'(x) = 6x + 5
Break it apart. Differentiate each term. Done.
The Product Rule
When two functions multiply, you can't just multiply the derivatives. Use:
(fg)' = f'g + fg'
Say it out loud: "first times derivative of second, plus second times derivative of first."
Example
f(x) = x²·sin(x)
Let f = x² and g = sin(x)
f' = 2x, g' = cos(x)
f'(x) = 2x·sin(x) + x²·cos(x)
This is the only correct approach. Don't try to distribute and differentiate—that method fails.
The Quotient Rule
When dividing functions, use:
(f/g)' = (f'g - fg') / g²
Mnemonic: "low d-high minus high d-low, over low-squared." Where "high" is the numerator function.
Example
f(x) = x² / (x + 1)
Let f = x², g = x + 1
f' = 2x, g' = 1
f'(x) = [(2x)(x+1) - (x²)(1)] / (x+1)²
f'(x) = (2x² + 2x - x²) / (x+1)²
f'(x) = (x² + 2x) / (x+1)²
Simplify where possible. Often you can factor and cancel.
Simpler Alternative
Sometimes rewriting the quotient as a product is easier. If the denominator is manageable, convert it:
f(x) = x² / (x + 1) = x²·(x+1)⁻¹
Then apply the product rule with the chain rule. Do whichever feels less error-prone.
The Chain Rule
This handles composite functions—functions within functions. If y = f(g(x)), then:
dy/dx = f'(g(x)) · g'(x)
Derivative of the outer function evaluated at the inner, times the derivative of the inner function.
Example
f(x) = (3x + 5)⁴
Outer function: u⁴, derivative is 4u³
Inner function: u = 3x + 5, derivative is 3
f'(x) = 4(3x + 5)³ · 3 = 12(3x + 5)³
Identify the inside and outside. Apply separately. Multiply.
Pattern Recognition
The chain rule appears constantly. Every time you see something raised to a power, you're looking at a composite function. The power rule alone won't cut it.
- f(x) = (x² + 3)⁵ → chain rule required
- f(x) = x⁵ → just power rule
Ask yourself: is the argument just x, or something more complicated? If more complicated, chain rule applies.
Trigonometric Derivatives
Memorize these. They come up constantly.
- d/dx[sin(x)] = cos(x)
- d/dx[cos(x)] = -sin(x)
- d/dx[tan(x)] = sec²(x)
- d/dx[csc(x)] = -csc(x)cot(x)
- d/dx[sec(x)] = sec(x)tan(x)
- d/dx[cot(x)] = -csc²(x)
Example
f(x) = sin(5x²)
Outer: sin(u), derivative is cos(u)
Inner: u = 5x², derivative is 10x
f'(x) = cos(5x²) · 10x = 10x·cos(5x²)
Chain rule applies to every trig function that has something other than x inside.
Exponential and Logarithmic Derivatives
Exponential:
- d/dx[eˣ] = eˣ
- d/dx[aˣ] = aˣ·ln(a)
eˣ is unique—it differentiates to itself. That's why it shows up everywhere.
Logarithmic:
- d/dx[ln(x)] = 1/x
- d/dx[logₐ(x)] = 1/(x·ln(a))
Example
f(x) = e^(3x²)
Outer: e^u, derivative is e^u
Inner: 3x², derivative is 6x
f'(x) = e^(3x²) · 6x = 6x·e^(3x²)
Common Derivatives Table
| Function | Derivative |
|---|---|
| c (constant) | 0 |
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| aˣ | aˣ·ln(a) |
| ln(x) | 1/x |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | sec²(x) |
| sec(x) | sec(x)tan(x) |
| csc(x) | -csc(x)cot(x) |
| cot(x) | -csc²(x) |
| arcsin(x) | 1/√(1-x²) |
| arccos(x) | -1/√(1-x²) |
| arctan(x) | 1/(1+x²) |
Higher-Order Derivatives
The second derivative is the derivative of the derivative: f''(x). Third is f'''(x), and so on.
Example: f(x) = 3x⁴
- f'(x) = 12x³
- f''(x) = 36x²
- f'''(x) = 72x
- f⁽⁴⁾(x) = 72
- f⁽⁵⁾(x) = 0
Each differentiation drops the power by one. After enough steps, you reach zero.
How to Differentiate Any Function
Follow this decision process:
- Is it a constant? → derivative is 0
- Is it x raised to a power? → power rule
- Is it a sum or difference? → differentiate term by term
- Is it a product of two functions? → product rule
- Is it a quotient? → quotient rule or rewrite as product
- Is there a function inside a function? → chain rule
- Is it a special function? → use the appropriate formula
Most functions require multiple rules in combination. A single derivative might need the product rule, chain rule, and trig derivatives all at once.
Worked Example
Find f'(x) if f(x) = 3x²·sin(4x)
Step 1: Identify the structure. Product of x² term and sin(4x).
Step 2: Apply product rule: (fg)' = f'g + fg'
Let u = 3x², v = sin(4x)
Step 3: Find u' = 6x (power rule)
Step 4: Find v'. This is composite: sin(u) where u = 4x.
v' = cos(4x) · 4 = 4cos(4x) (chain rule)
Step 5: Combine:
f'(x) = (6x)·sin(4x) + (3x²)·(4cos(4x))
f'(x) = 6x·sin(4x) + 12x²·cos(4x)
That's the answer. Break it into steps. Don't try to do everything mentally at once.
Common Mistakes
- Forgetting the chain rule. If something is inside another function, you need it. Every time.
- Applying power rule to everything. Doesn't work for products or quotients.
- Sign errors in quotient rule. Remember: f'g - fg'. The subtraction is easy to drop.
- Simplifying prematurely. Get the derivative correct first. Simplify after.
- Confusing product and chain. Product is for fg. Chain is for f(g(x)). Different situations.
Implicit Differentiation (Briefly)
When y isn't isolated, differentiate both sides with respect to x, treating y as a function of x. Every time you differentiate y, multiply by dy/dx.
Example: x² + y² = 25
2x + 2y·(dy/dx) = 0
2y·(dy/dx) = -2x
dy/dx = -x/y
You get dy/dx in terms of both x and y. That's fine. That's how implicit differentiation works.