What a derivative actually measures
A derivative describes how fast a function's output changes relative to its input at a specific point — geometrically, the slope of the tangent line to the function's graph at that point. Everything else in differential calculus builds on mechanically computing this rate of change for increasingly complex functions, which is exactly what a fixed set of rules lets you do without recomputing a limit from scratch every time.
The power rule: the one everyone learns first
For f(x) = x^n, the derivative is f'(x) = n·x^(n-1) — multiply by the exponent, then reduce the exponent by one.
f(x) = x^3 → f'(x) = 3x²
f(x) = x^0.5 → f'(x) = 0.5x^(-0.5)
This rule only applies directly to a constant numeric exponent — x^x needs a different technique (logarithmic differentiation) not covered by the plain power rule.
The product rule: differentiating f(x)·g(x)
(f·g)' = f'·g + f·g' — differentiate each factor in turn, multiply by the other factor unchanged, and add the two results. This is necessary because the derivative of a product isn't simply the product of the derivatives — (x²·sin(x))' is not 2x·cos(x), it's 2x·sin(x) + x²·cos(x).
The quotient rule: differentiating f(x)/g(x)
(f/g)' = (f'·g - f·g') / g² — a variation on the product rule accounting for the denominator changing too. Note the subtraction and the order matters: swapping f'g and fg' gives the wrong sign entirely.
The chain rule: differentiating a function of a function
(f(g(x)))' = f'(g(x)) · g'(x) — differentiate the outer function (leaving the inner function's argument untouched), then multiply by the derivative of the inner function. This is the rule that makes sin(x²)'s derivative cos(x²) · 2x, not just cos(x²) — the 2x factor (the derivative of the inner x²) is easy to forget, and forgetting it is the single most common differentiation mistake.
Why these four rules cover almost everything
Every one of these rules reduces a complex expression to simpler sub-expressions, recursively — a symbolic differentiator applies the matching rule at each level of an expression's structure (is this a sum? apply the sum rule to each term. A product? Apply the product rule. A function call like sin(...)? Apply the chain rule with that specific function's known derivative) until it bottoms out at the simplest pieces: constants (derivative 0) and x itself (derivative 1).
| Expression shape | Rule applied |
|---|---|
x^n |
Power rule |
f(x) · g(x) |
Product rule |
f(x) / g(x) |
Quotient rule |
f(g(x)) (a function applied to a sub-expression) |
Chain rule |
f(x) + g(x) |
Sum rule (differentiate each term separately) |
Common mistakes
- Forgetting the chain rule's extra factor.
sin(x²)'s derivative needs the2xfrom differentiating the innerx²— dropping it is the most common calculus mistake at this level. - Getting the quotient rule's subtraction order backward.
f'g - fg', notfg' - f'g— the sign flips the entire result. - Applying the power rule to a variable exponent.
x^xisn'tx·x^(x-1)— that rule only holds for constant exponents.
FAQ
Why does the chain rule matter so much in practice?
Most real functions are compositions — a function applied to another expression, not just x alone — so nearly every non-trivial derivative involves the chain rule at some level.
Is there a single rule that covers every possible function?
No — these four rules (plus the sum rule) combine recursively to cover the vast majority of elementary functions, but some exotic cases (implicit differentiation, variable exponents) need additional techniques beyond this basic set.
Why does differentiating x^x need a different technique?
The power rule assumes a constant exponent; when the exponent itself depends on x, the expression needs logarithmic differentiation (taking the natural log of both sides first) instead.
Compute the symbolic derivative of any function and evaluate it at a point with the Derivative Calculator — entirely in your browser.