How Differentiation Works
A derivative measures a function's instantaneous rate of change — the slope of its curve at any point. Differentiation follows a small set of rules, and nearly every problem is solved by recognizing which rules apply and applying them in the right order.
The Core Rules
- Power Rule: d/dx[xⁿ] = n·xⁿ⁻¹. The workhorse — try it on x³ to get 3x².
- Sum Rule: differentiate term by term; constants vanish.
- Product Rule: (uv)′ = u′v + uv′ for multiplied functions.
- Quotient Rule: (u/v)′ = (u′v − uv′) / v² for divisions.
- Chain Rule: for nested functions like sin(x²) — outer derivative times inner derivative.
A Worked Example
Differentiate f(x) = x²·eˣ. The product rule gives (2x)(eˣ) + (x²)(eˣ) = eˣ(2x + x²). Notice the strategy: identify the structure first (a product), then differentiate each piece with the power and exponential rules. The calculator above follows the same logic and names the rules it uses.
Need to evaluate the derivative at a point? Plug values into our scientific calculator.