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Derivative Rules Calculator

Differentiation-rules demo. Enter f, g and x to numerically compute f′, g′, then verify sum, product and quotient rules—for learning the basic rules.

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(fg)′=f′g+fg′ (f/g)′=(f′g−fg′)/g²
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📖 Tutorial｜Derivative Rules

1. Definition
Sum/difference (f±g)′=f′±g′; product (fg)′=f′g+fg′; quotient (f/g)′=(f′g−fg′)/g².
2. Symbols
RuleFormula
Sum/diff(f±g)′=f′±g′
Product(fg)′=f′g+fg′
Quotient(f/g)′=(f′g−fg′)/g²
Constant(cf)′=cf′
3. How it works
  • We numerically compute f′ and g′ by central differences;
  • Then plug them into each rule;
  • Each step is shown numerically.
4. Steps
  1. Enter f(x) and g(x);
  2. Enter the point x;
  3. Click Calculate to see each rule numerically.
5. Example
Example: f=x², g=x at x=2. f′=4, g′=1, so (fg)′=4·2+4·1=12.
6. Pitfalls
Product rule is f′g+fg′, not f′g′;
In the quotient rule the numerator is f′g−fg′—keep the order;
The quotient rule fails where g=0.

❓ FAQ｜Derivative Rules

Why is the product rule not f′g′?
The product changes for two reasons (f changes and g changes), so both terms must be added.
How to remember the quotient rule?
“lo d hi minus hi d lo, over lo squared”: (f′g−fg′)/g².
What is the derivative of a constant?
Zero, so (cf)′=cf′.
Can I differentiate a sum term by term?
Yes. (f+g)′=f′+g′.
How accurate is the number?
Central differences give error around 1e-5, enough for textbook examples.