Example 1, Power rule
d/dx(x³)
3x²
- 1
Apply power rule
n = 3, so d/dx(x³) = 3·x³⁻¹ - 2
Simplify
= 3x²
Power rule: d/dx(xⁿ) = n·xⁿ⁻¹.
Power · Chain · Product · Quotient
Derivative Calculator d/dx with Steps
Find the derivative of any function using the power, chain, product, and quotient rules. Worked examples walk you through each technique step by step.
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The rules of differentiation
The Four Differentiation Rules
(xⁿ)' = nxⁿ⁻¹ · (fg)' = f'g + fg' · (f(g))' = f'(g) · g'
Power, product, and chain. With the quotient rule, these four cover almost every derivative you'll meet.
The derivative measures the rate of change of a function. The five core rules cover almost every problem you'll meet: power rule (xⁿ → nxⁿ⁻¹), sum rule (linearity), product rule, quotient rule, and chain rule. Combined with the standard derivatives of trig, exponential, and log functions, these rules let you differentiate anything you can write down.
The most error-prone is the chain rule, when one function is inside another, multiply by the derivative of the inner part. Forgetting this is the most common calculus mistake.
Worked Examples
Five derivatives, every rule covered
Power rule, sum rule, chain rule, product rule, quotient rule, one example each.
Example 2, Sum and constant rule
d/dx(2x³ + 5x − 7)
6x² + 5
- 1
Differentiate term by term
d/dx(2x³) + d/dx(5x) + d/dx(−7) - 2
Apply each rule
6x² + 5 + 0 - 3
Result
6x² + 5
Derivative of a sum = sum of derivatives. Constants vanish.
Example 3, Chain rule
d/dx(sin(x²))
2x·cos(x²)
- 1
Identify outer and inner
Outer = sin(u), inner = u = x² - 2
Derivatives
d/du(sin(u)) = cos(u), d/dx(x²) = 2x - 3
Multiply
cos(x²) · 2x = 2x·cos(x²)
Chain rule: derivative of outer × derivative of inner.
Example 4, Product rule
d/dx(x²·sin(x))
2x·sin(x) + x²·cos(x)
- 1
Identify
f = x², g = sin(x) - 2
Derivatives
f' = 2x, g' = cos(x) - 3
Apply formula
2x · sin(x) + x² · cos(x)
Product rule: (fg)' = f'g + fg'.
Example 5, Quotient rule
d/dx(x / (x² + 1))
(1 − x²) / (x² + 1)²
- 1
Identify
f = x, g = x² + 1 - 2
Derivatives
f' = 1, g' = 2x - 3
Apply formula
(1·(x² + 1) − x·2x) / (x² + 1)² - 4
Simplify
= (x² + 1 − 2x²) / (x² + 1)² = (1 − x²) / (x² + 1)²
Quotient rule: (f/g)' = (f'g − fg') / g².
Step-by-step derivative calculator
Derivatives are the heart of calculus, they measure slope, velocity, growth rate, marginal cost, and anything else that changes. This page covers the standard differentiation rules with worked examples for each. The calculator above evaluates numerical expressions for verification, the examples walk through symbolic differentiation by hand.
Power rule and basic derivatives
The power rule, d/dx(xⁿ) = nxⁿ⁻¹, applies to any real exponent n. Combined with the sum rule (derivatives distribute over addition) and the constant multiple rule (constants pass through), this handles all polynomial differentiation. Standard derivatives like d/dx(sin x) = cos x, d/dx(eˣ) = eˣ, and d/dx(ln x) = 1/x round out the toolkit.
Chain rule, the most important rule
If a function is built from another function (composition), the chain rule says: differentiate the outer, evaluate at the inner, then multiply by the derivative of the inner. For sin(x²), the outer is sin and inner is x², giving cos(x²)·2x. Almost every nontrivial calculus problem uses chain rule somewhere.
Product and quotient rules
The product rule (fg)' = f'g + fg' is needed when neither factor is constant. The quotient rule (f/g)' = (f'g − fg')/g² applies to ratios. Both can be derived from chain rule but are worth memorizing as separate formulas because they appear so often.
When and how to use this page
Plug numerical values into the calculator to verify that a specific function's derivative gives the slope you expect. The worked examples below show the full algebraic derivation for the five most important differentiation patterns. For symbolic derivatives of arbitrary functions, use the photo solver.
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