Derivative Rules — Formulas, Table & Examples

Derivative Rules

A list of common derivative rules is given below.

Four basic derivative rules: constant (d/dx of α = 0), scalar multiple, sum (d/dx(u+v) = du/dx + dv/dx), and difference.

Five derivative rules: Power, Product, Quotient, Reciprocal, and Chain, shown with formulas using standard calculus notation...

Trig and inverse trig derivative rules: d/dx sin u, cos u, tan u, cot u, sec u, csc u, and arcsin, arccos, arctan, arccot,...

Derivative rules 12–14: exponential d/dx(e^u)=e^u·du/dx and d/dx(a^u)=a^u·ln(a)·du/dx; log d/dx(ln u)=(1/u)·du/dx and...

Key Formula

\begin{aligned} &\textbf{Constant:}\quad \frac{d}{dx}[c] = 0 \\ &\textbf{Power:}\quad \frac{d}{dx}[x^n] = n\,x^{n-1} \\ &\textbf{Constant Multiple:}\quad \frac{d}{dx}[c\,f(x)] = c\,f'(x) \\ &\textbf{Sum/Difference:}\quad \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) \\ &\textbf{Product:}\quad \frac{d}{dx}[f(x)\,g(x)] = f'(x)\,g(x) + f(x)\,g'(x) \\ &\textbf{Quotient:}\quad \frac{d}{dx}\!\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{[g(x)]^2} \\ &\textbf{Chain:}\quad \frac{d}{dx}[f(g(x))] = f'(g(x))\cdot g'(x) \end{aligned}

Where:

$c$ = A constant (a fixed real number)
$n$ = Any real-number exponent
$f(x), g(x)$ = Differentiable functions of x
$f'(x), g'(x)$ = The derivatives of f and g with respect to x

Worked Example

Problem: Find the derivative of f(x) = 3x⁴ + 5x² − 7x + 2.

Step 1: Apply the sum/difference rule: differentiate each term separately.

f'(x) = \frac{d}{dx}[3x^4] + \frac{d}{dx}[5x^2] - \frac{d}{dx}[7x] + \frac{d}{dx}[2]

Step 2: Use the constant multiple rule and power rule on each term. For 3x⁴: bring down the exponent 4 and reduce it by 1.

\frac{d}{dx}[3x^4] = 3 \cdot 4x^{3} = 12x^3

Step 3: Differentiate the remaining terms the same way. For 5x²: exponent 2 comes down. For 7x: this is 7x¹. For the constant 2: the derivative is 0.

\frac{d}{dx}[5x^2] = 10x, \quad \frac{d}{dx}[7x] = 7, \quad \frac{d}{dx}[2] = 0

Step 4: Combine all the results.

f'(x) = 12x^3 + 10x - 7

Answer: f'(x) = 12x³ + 10x − 7

Another Example

This example requires the chain rule for a composite function, whereas the first example only used the power rule and sum rule on a polynomial.

Problem: Find the derivative of h(x) = (2x + 1)⁵.

Step 1: Identify this as a composite function. The outer function is u⁵ and the inner function is u = 2x + 1. The chain rule is needed.

h(x) = f(g(x)) \quad\text{where}\quad f(u) = u^5,\; g(x) = 2x+1

Step 2: Differentiate the outer function using the power rule, keeping the inner function in place.

f'(g(x)) = 5(2x+1)^4

Step 3: Differentiate the inner function.

g'(x) = 2

Step 4: Multiply the two results together (chain rule).

h'(x) = 5(2x+1)^4 \cdot 2 = 10(2x+1)^4

Answer: h'(x) = 10(2x + 1)⁴

Frequently Asked Questions

What is the difference between the product rule and the chain rule?

The product rule applies when two separate functions are multiplied together, such as f(x)·g(x). The chain rule applies when one function is nested inside another, such as f(g(x)). A good test: if you can split the expression into two factors that both depend on x, use the product rule. If one function feeds into another (a 'function of a function'), use the chain rule. Sometimes you need both in the same problem.

How do you know which derivative rule to use?

Look at the structure of the expression. If terms are added or subtracted, use the sum/difference rule to handle each term on its own. If two functions are multiplied, use the product rule. If one is divided by another, use the quotient rule. If a function is composed inside another, use the chain rule. For any single term like xⁿ, the power rule is your go-to tool.

Do derivative rules work for negative and fractional exponents?

Yes. The power rule works for any real exponent n, including negative numbers and fractions. For example, the derivative of x^(−2) is −2x^(−3), and the derivative of x^(1/2) is (1/2)x^(−1/2). This makes the power rule extremely versatile, since square roots, cube roots, and reciprocals can all be rewritten with fractional or negative exponents.

Derivative Rules vs. Integral Rules

	Derivative Rules	Integral Rules
Purpose	Find the rate of change (slope) of a function	Find the antiderivative or accumulated area under a curve
Power formula	d/dx [xⁿ] = n·xⁿ⁻¹	∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
Direction	Breaks a function down (differentiation)	Builds a function up (antidifferentiation)
Uniqueness of result	The derivative of a function is unique	Antiderivatives include an arbitrary constant C

Why It Matters

Derivative rules form the backbone of every calculus course. You will use them constantly — to find slopes of tangent lines, determine where functions increase or decrease, solve optimization problems, and analyze motion. Mastering these rules also prepares you for integral calculus, since many integration techniques rely on recognizing derivative patterns in reverse.

Common Mistakes

Mistake: Applying the power rule to a composite function without the chain rule. For example, writing the derivative of (3x + 1)⁴ as 4(3x + 1)³ instead of 12(3x + 1)³.

Correction: Whenever the base is more than just x, you must also multiply by the derivative of the inner function. Here the inner derivative is 3, giving 4(3x + 1)³ · 3 = 12(3x + 1)³.

Mistake: Confusing the product rule with simply multiplying the individual derivatives: writing d/dx[f·g] = f'·g' instead of f'·g + f·g'.

Correction: The product rule requires two terms: the derivative of the first times the second, plus the first times the derivative of the second. There is no shortcut that lets you just multiply the derivatives together.

Related Terms

Derivative — The core concept these rules compute
Power Rule — Most frequently used single derivative rule
Product Rule — Rule for differentiating products of functions
Quotient Rule — Rule for differentiating ratios of functions
Chain Rule — Rule for differentiating composite functions
Implicit Differentiation — Applies derivative rules to implicitly defined curves
Logarithmic Differentiation — Technique using logs to simplify complex derivatives
Integral Rules — The reverse process — antidifferentiation formulas