Derivative Rules- Essential Calculus Formulas Explained
What Derivatives Actually Are (No Fluff)
A derivative tells you the instantaneous rate of change at any point on a curve. That's it. Nothing magical, nothing abstract. You take a function, apply the right rule, and you get another function that shows you how fast the original one is changing.
If you're taking calculus, you need these rules memorized. Not "kind of" memorized. Actually memorized. The problems won't wait for you to look them up.
The Basic Building Blocks
Power Rule
This is the one you use most. If you have xⁿ, the derivative is n·xⁿ⁻¹.
Examples:
- d/dx(x³) = 3x²
- d/dx(x⁵) = 5x⁴
- d/dx(x²) = 2x
Works for any real exponent, positive, negative, or fractional. Drop the exponent in front, subtract one from the exponent.
Constant Rule
The derivative of any constant is zero. That's because a flat line has zero slope.
- d/dx(5) = 0
- d/dx(-12) = 0
- d/dx(π) = 0
Constant Multiple Rule
Constants just float through differentiation. If you have c·f(x), the derivative is c·f'(x).
- d/dx(4x³) = 4·3x² = 12x²
- d/dx(½x⁴) = ½·4x³ = 2x³
Sum and Difference Rules
Derivative of a sum = sum of derivatives. Derivative of a difference = difference of derivatives. Just differentiate each term separately.
d/dx(x² + 3x - 7) = 2x + 3
The Rules That Actually Trip People Up
Product Rule
When two functions multiply, you can't just multiply their derivatives. Use this formula:
d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Think "first times derivative of second, plus second times derivative of first."
Example: d/dx(x²·sin x)
- f(x) = x², so f'(x) = 2x
- g(x) = sin x, so g'(x) = cos x
- Answer: 2x·sin x + x²·cos x
Quotient Rule
When dividing functions, the formula is:
d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²
Remember: "low d-high minus high d-low, over low squared."
Example: d/dx(x/(x+1))
- f(x) = x, f'(x) = 1
- g(x) = x+1, g'(x) = 1
- Answer: [(1)(x+1) - (x)(1)]/(x+1)² = 1/(x+1)²
Chain Rule
This is for composite functions — one function inside another. If y = f(g(x)), then:
dy/dx = f'(g(x)) · g'(x)
Take the derivative of the outer function, keep the inner function, multiply by the derivative of the inner function.
Example: d/dx(sin(x²))
- Outer: sin(u), derivative is cos(u)
- Inner: u = x², derivative is 2x
- Answer: cos(x²) · 2x = 2x·cos(x²)
The chain rule shows up everywhere. Powers, roots, trigonometric functions, exponential functions — if there's a function inside a function, you're using the chain rule.
Trigonometric Derivatives
These come up constantly. Memorize them now:
- 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
The negative sign on cosine and cosecant trips people up. Watch out for it.
Exponential and Logarithmic Derivatives
The derivative of eˣ is eˣ. That's the whole reason e is special.
- d/dx(eˣ) = eˣ
- d/dx(eᵘ) = eᵘ·u' (chain rule applies)
- d/dx(aˣ) = aˣ·ln(a)
For logarithms:
- d/dx(ln x) = 1/x
- d/dx(logₐx) = 1/(x·ln a)
- d/dx(ln u) = u'/u
The derivative of ln(u) is u'/u. This shows up constantly in calculus problems.
Common Derivative Formulas
| Function | Derivative |
|---|---|
| xⁿ | n·xⁿ⁻¹ |
| sin x | cos x |
| cos x | -sin x |
| eˣ | eˣ |
| ln x | 1/x |
| aˣ | aˣ·ln(a) |
| tan x | sec² x |
| cot x | -csc² x |
| sec x | sec x·tan x |
| csc x | -csc x·cot x |
How to Actually Use These Rules
Step 1: Identify the Structure
Look at your function. Is it a power of x? A sum? A product? A quotient? A composition? This determines which rule you use.
Step 2: Apply the Right Rule
Once you know the structure:
- Simple power? Power rule.
- Two things multiplied? Product rule.
- One thing divided by another? Quotient rule.
- Something inside something else? Chain rule.
Step 3: Simplify
Clean up your answer. Combine like terms. Reduce fractions. Your teacher will mark you down for messy answers.
Example Problem
Find the derivative of: f(x) = 3x²·cos(2x)
This is a product of 3x² and cos(2x). Use the product rule.
- u = 3x², u' = 6x
- v = cos(2x), v' = -sin(2x)·2 = -2sin(2x)
- f'(x) = (6x)·cos(2x) + (3x²)·(-2sin(2x))
- f'(x) = 6x·cos(2x) - 6x²·sin(2x)
- f'(x) = 6x[cos(2x) - x·sin(2x)]
Common Mistakes That Cost Points
- Forgetting the chain rule on composite functions. If you see parentheses or nested functions, you're probably chain-ruling.
- Dropping the constant in the product rule or quotient rule formula. Both terms are always there.
- Sign errors in the quotient rule (it's subtraction, not addition) and trig derivatives (cosine and cosecant have negatives).
- Forgetting to simplify. Raw derivatives are often messier than they need to be.
These rules aren't suggestions. They're the actual mechanics of calculus. Every problem you encounter in differential calculus is some combination of these rules applied to different functions.
Study them. Know them cold.