Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools


Derivative Rules



Constant Rule
Power Rule
Constant Multiple Rule
Sum and Difference Rules
Product Rule
Quotient Rule
Chain Rule
Mean Value Theorem
Rolle's Theorem
L'Hôpital's Rule
All Rules and Theorems at a Glance



Replacing the Limit with Algebra


The limit definition of the derivative is precise but impractical for routine computation. Evaluating lim⁡h→0f(a+h)−f(a)h\lim_{h \to 0} \frac{f(a+h) - f(a)}{h} for every function and every point would make calculus unworkable. Differentiation rules extract patterns from the definition and package them into algebraic formulas that apply to entire classes of functions.

The basic rules—power, sum, product, quotient, chain—handle nearly all explicit functions encountered in practice. Each rule is proven once from the limit definition, then used without returning to it. Together they reduce differentiation to mechanical manipulation, freeing attention for the harder question of what the derivative means in context.

Beyond computational rules, three theorems govern the behavior of derivatives on intervals. The Mean Value Theorem connects instantaneous and average rates of change. Rolle's Theorem guarantees horizontal tangents under symmetric conditions. L'Hôpital's rule turns indeterminate limits into derivative computations.

Key Terms

Derivative— the object these rules compute
Differentiability— the hypothesis required for every rule to apply

See All Calculus Definitions →


Power Rule


For any real exponent nn:

ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}


The rule covers positive integers (x3→3x2x^3 \to 3x^2), negative integers (x−2→−2x−3x^{-2} \to -2x^{-3}), fractions (x1/2→12x−1/2x^{1/2} \to \frac{1}{2}x^{-1/2}), and irrational exponents (xπ→πxπ−1x^\pi \to \pi x^{\pi - 1}). For positive integer nn, the proof uses the binomial expansion of (x+h)n(x+h)^n in the limit definition. Extension to rational and real exponents requires logarithmic differentiation or the generalized exponential xn=enln⁡xx^n = e^{n \ln x}.

The power rule is the most frequently applied differentiation formula. Combined with the constant multiple and sum rules, it handles all polynomials and many algebraic expressions directly.

Constant Multiple Rule


If cc is a constant and ff is differentiable, then

ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)


Constants factor out of derivatives. The proof is immediate from the limit definition: the constant cc factors out of the difference quotient and passes through the limit.

This rule is a special case of the product rule (one factor being constant), but it is worth stating separately. In practice, factoring constants out before differentiating simplifies every computation.

Sum and Difference Rules


If ff and gg are both differentiable, then

ddx[f(x)+g(x)]=f′(x)+g′(x)\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)


ddx[f(x)−g(x)]=f′(x)−g′(x)\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)


The derivative of a sum is the sum of the derivatives. This follows from the limit definition by splitting the difference quotient into two parts and applying the sum rule for limits.

The rule extends to any finite number of terms: the derivative of f1+f2+⋯+fnf_1 + f_2 + \cdots + f_n is f1′+f2′+⋯+fn′f_1' + f_2' + \cdots + f_n'. Combined with the constant multiple rule, this handles all linear combinations. In particular, every polynomial is differentiated term by term.

Product Rule


If ff and gg are both differentiable, then

ddx[f(x)⋅g(x)]=f′(x)⋅g(x)+f(x)⋅g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)


The derivative of a product is not the product of the derivatives. Each factor takes a turn being differentiated while the other remains unchanged, and the two contributions are added.

The proof adds and subtracts f(x+h)g(x)f(x+h)g(x) inside the difference quotient for f(x)g(x)f(x)g(x), splitting it into a piece that isolates g′g' and a piece that isolates f′f'. The fact that differentiability implies continuity ensures that f(x+h)→f(x)f(x+h) \to f(x) as h→0h \to 0.

For three factors: (fgh)′=f′gh+fg′h+fgh′(fgh)' = f'gh + fg'h + fgh'. The pattern generalizes—each factor is differentiated once while all others remain, and the results are summed.

Quotient Rule


If ff and gg are differentiable and g(x)≠0g(x) \neq 0, then

ddx[f(x)g(x)]=f′(x)⋅g(x)−f(x)⋅g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}


The numerator subtracts where the product rule adds, and the entire expression is divided by the square of the denominator. The order matters: f′g−fg′f'g - fg', not fg′−f′gfg' - f'g.

The quotient rule can be derived from the product rule by writing f/g=f⋅g−1f/g = f \cdot g^{-1} and applying the chain rule to g−1g^{-1}. Both approaches yield the same formula. In practice, the quotient rule is applied directly when the function is naturally a fraction, and rewriting as a product is preferred when the denominator is a simple power.

Chain Rule


If gg is differentiable at xx and ff is differentiable at g(x)g(x), then the composite function f∘gf \circ g is differentiable at xx and

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)


Differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function. In Leibniz notation, with y=f(u)y = f(u) and u=g(x)u = g(x):

dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}


The notation suggests fraction cancellation, and while dy/dxdy/dx is not literally a fraction, the chain rule makes Leibniz notation behave as though it were.

The chain rule extends to deeper compositions. For f(g(h(x)))f(g(h(x))): differentiate the outermost function at its argument, multiply by the derivative of the next layer at its argument, and continue inward. Each layer contributes a multiplicative factor.

This is the most powerful of the basic rules. Without it, composite functions—sin⁡(x2)\sin(x^2), e3xe^{3x}, ln⁡(cos⁡x)\ln(\cos x)—cannot be differentiated. Most applications of differentiation involve the chain rule at some level.

Mean Value Theorem


If ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there exists at least one point c∈(a,b)c \in (a, b) such that

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}


The instantaneous rate of change at cc equals the average rate of change over the entire interval. Geometrically, there is a point where the tangent line is parallel to the secant line connecting (a,f(a))(a, f(a)) and (b,f(b))(b, f(b)).

The Mean Value Theorem is primarily a theoretical tool. It proves that a function with positive derivative on an interval is increasing. It establishes that two functions with the same derivative differ by a constant. It justifies the connection between antiderivatives and definite integrals in the Fundamental Theorem of Calculus.

The hypotheses are essential. Continuity on the closed interval [a,b][a, b] ensures no gaps at the endpoints. Differentiability on the open interval (a,b)(a, b) ensures the derivative exists at interior points. Functions that violate either condition may fail the conclusion.

Rolle's Theorem


Rolle's Theorem is the special case of the Mean Value Theorem where f(a)=f(b)f(a) = f(b).

If ff is continuous on [a,b][a, b], differentiable on (a,b)(a, b), and f(a)=f(b)f(a) = f(b), then there exists at least one point c∈(a,b)c \in (a, b) with f′(c)=0f'(c) = 0.

A function that starts and ends at the same value must have a horizontal tangent somewhere between. If the function rises above its starting value, it must turn around—creating a maximum with zero slope. If it drops below, it must turn upward—creating a minimum. If it stays constant, f′(x)=0f'(x) = 0 everywhere on the interval.

Rolle's Theorem is often used as a stepping stone in proofs. The Mean Value Theorem itself is proven by applying Rolle's Theorem to an auxiliary function constructed to satisfy g(a)=g(b)g(a) = g(b). It also appears in arguments about the number of roots: between any two roots of ff, there must be a root of f′f'.

L'Hôpital's Rule


If lim⁡x→af(x)g(x)\lim_{x \to a} \frac{f(x)}{g(x)} produces the indeterminate form 00\frac{0}{0} or ∞∞\frac{\infty}{\infty}, and if lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f'(x)}{g'(x)} exists (or equals ±∞\pm\infty), then

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}


The rule replaces a difficult limit of functions with a (potentially simpler) limit of their derivatives. It applies equally when aa is finite or a=±∞a = \pm\infty.

Critical requirements: the original limit must be an indeterminate form, and the limit of the derivative ratio must exist or be ±∞\pm\infty. If the derivative ratio produces another indeterminate form, L'Hôpital's rule may be applied again. If the derivative ratio oscillates or fails to exist, the rule gives no information—it does not say the original limit fails to exist, only that this method does not resolve it.

Other indeterminate forms—0⋅∞0 \cdot \infty, ∞−∞\infty - \infty, 000^0, 1∞1^\infty, ∞0\infty^0—can be converted to 00\frac{0}{0} or ∞∞\frac{\infty}{\infty} through algebraic rearrangement, then handled by L'Hôpital's rule. The conversion step varies by form: 0⋅∞0 \cdot \infty becomes a fraction by moving one factor to the denominator; exponential forms use the identity f(x)g(x)=eg(x)ln⁡f(x)f(x)^{g(x)} = e^{g(x) \ln f(x)}.

The Mean Value Theorem, Rolle's Theorem, and L'Hôpital's rule are the three theorems on this page that govern derivative behavior on intervals. The table below sets them side by side across their hypotheses, conclusions, and the role each plays in practice.
Theorem Hypotheses Conclusion Primary use
Mean Value Theorem f continuous on [a, b]; f differentiable on (a, b) there is some c ∈ (a, b) with f'(c) = (f(b) − f(a)) ⁄ (b − a) connecting instantaneous and average rates; foundation for many subsequent results
Rolle's Theorem f continuous on [a, b]; f differentiable on (a, b);  f(a) = f(b) there is some c ∈ (a, b) with f'(c) = 0 stepping stone in proofs (including MVT itself); counting roots via f'
L'Hôpital's Rule limit of f ⁄ g gives indeterminate form 0 ⁄ 0 or ∞ ⁄ ∞; limit of f' ⁄ g' exists (or is ±∞) lim f ⁄ g = lim f' ⁄ g' resolving indeterminate limits via differentiation

All Rules and Theorems at a Glance


The ten formulas above split naturally into seven algebraic rules — constant, constant multiple, power, sum/difference, product, quotient, chain — and three interval theorems — Mean Value, Rolle's, and L'Hôpital's. The table below collects every formula introduced on this page as a single reference card, with the key condition or note that governs its use.
Rule or theorem Formula Note
Constant rule if f(x) = c then f'(x) = 0 constant functions have zero slope everywhere
Constant multiple rule (c · f(x))' = c · f'(x) constants factor straight through differentiation
Power rule (xn)' = n · xn − 1 valid for any real exponent n
Sum & difference rule (f ± g)' = f' ± g' extends to any finite number of terms; differentiate term by term
Product rule (f · g)' = f' · g + f · g' each factor takes a turn being differentiated; generalises to more than two factors
Quotient rule (f ⁄ g)' = (f' · g − f · g') ⁄ g2 requires g(x) ≠ 0; numerator order matters
Chain rule (f(g(x)))' = f'(g(x)) · g'(x) handles compositions; extends to any depth of nesting
Mean Value Theorem f'(c) = (f(b) − f(a)) ⁄ (b − a) for some c ∈ (a, b) requires continuity on [a, b] and differentiability on (a, b)
Rolle's Theorem f'(c) = 0 for some c ∈ (a, b) special case of MVT when f(a) = f(b)
L'Hôpital's Rule lim f ⁄ g = lim f' ⁄ g' applies to indeterminate forms 0 ⁄ 0 and ∞ ⁄ ∞; other forms must be converted first