Derivative Calculator - Complete Guide to Differentiation

You memorized the power rule. Now your professor throws x²·sin(x) at you and expects you to know what to do. That's where the product rule, quotient rule, and chain rule come in.

This guide covers every differentiation rule you need — with examples that actually make sense.

The Four Core Rules

1. Power Rule — The Foundation

d/dx(xⁿ) = n · xⁿ⁻¹

This handles polynomials. For f(x) = 3x⁴ − 2x² + 7:

f'(x) = 12x³ − 4x

Constants vanish. Exponents come down. Done.

2. Product Rule — Two Functions Multiplied

When f(x) = u(x) · v(x):

(uv)' = u'v + uv'

Translation: derivative of first × second + first × derivative of second.

Example: f(x) = x² · eˣ

• u = x², u' = 2x
• v = eˣ, v' = eˣ
• f'(x) = 2x·eˣ + x²·eˣ = eˣ(2x + x²)

3. Quotient Rule — One Function Divided by Another

When f(x) = u(x) / v(x):

(u/v)' = (u'v − uv') / v²

Example: f(x) = (3x + 1) / (x − 2)

• u = 3x + 1, u' = 3
• v = x − 2, v' = 1
• f'(x) = [3(x−2) − (3x+1)(1)] / (x−2)²
• f'(x) = (3x − 6 − 3x − 1) / (x−2)²
• f'(x) = −7 / (x−2)²

4. Chain Rule — Functions Inside Functions

When f(x) = g(h(x)):

f'(x) = g'(h(x)) · h'(x)

Example: f(x) = sin(3x²)

• Outer: sin(u) → cos(u)
• Inner: 3x² → 6x
• f'(x) = cos(3x²) · 6x = 6x·cos(3x²)

Which Rule Do I Use? Decision Tree

| Situation | Rule | Example | |-----------|------|---------| | Single polynomial term | Power Rule | x⁵ → 5x⁴ | | Two functions multiplied | Product Rule | x²·sin(x) | | One function divided by another | Quotient Rule | (x+1)/(x−1) | | Function inside a function | Chain Rule | e^(2x) → 2e^(2x) | | Multiple rules needed | Combine them | x²·sin(3x) |

The Trench Truth: The #1 mistake on derivative exams is using the product rule when the power rule alone suffices. For 3x⁴, just use the power rule — don't split it into 3 · x⁴ and apply the product rule. You'll get the same answer but waste time and invite arithmetic errors.

Common Derivatives Reference Table

| Function | Derivative | Function | Derivative | |----------|-----------|----------|-----------| | xⁿ | nxⁿ⁻¹ | eˣ | eˣ | | sin(x) | cos(x) | cos(x) | −sin(x) | | tan(x) | sec²(x) | ln(x) | 1/x | | logₐ(x) | 1/(x·ln a) | aˣ | aˣ·ln a |

Real-World Applications

• Physics: Velocity is the derivative of position. Acceleration is the derivative of velocity.
• Economics: Marginal cost = derivative of total cost. Marginal revenue = derivative of total revenue.
• Biology: Population growth rate = derivative of population over time.

Try our derivative calculator to verify your work on any polynomial.

Related: Integral Calculator · Quadratic Formula Calculator · Square Root Calculator