Derivative Rules

Constant Rule

Power Rule

Constant Multiple Rule

Sum and Difference Rules

All Rules and Theorems at a Glance

Derivative Rules FAQ

Replacing the Limit with Algebra

The limit definition of the derivative is precise but impractical for routine computation. Evaluating

\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

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Constant Rule

f(x) = c

for some constant

c

, then

Constant Rule (Derivatives)

\frac{d}{dx}[c] = 0

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A constant function has a horizontal graph—its slope is zero everywhere. From the limit definition: the difference quotient

\frac{c - c}{h} = 0

for all

h

, so the limit is

0

.

This rule is simple but essential. It establishes the baseline: quantities that do not change have zero rate of change. It also appears implicitly in every application of the sum rule, where constant terms vanish under differentiation.

Power Rule

For any real exponent

n

Power Rule (Derivatives)

\frac{d}{dx}[x^n] = n x^{n-1}

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The rule covers positive integers (

x^3 \to 3x^2

), negative integers (

x^{-2} \to -2x^{-3}

), fractions (

x^{1/2} \to \frac{1}{2}x^{-1/2}

), and irrational exponents (

x^\pi \to \pi x^{\pi - 1}

). For positive integer

n

, the proof uses the binomial expansion of

(x+h)^n

in the limit definition. Extension to rational and real exponents requires logarithmic differentiation or the generalized exponential

x^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

c

is a constant and

f

is differentiable, then

Constant Multiple Rule (Derivatives)

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

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Constants factor out of derivatives. The proof is immediate from the limit definition: the constant

c

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

f

and

g

are both differentiable, then

Sum and Difference Rule (Derivatives)

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

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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

f_1 + f_2 + \cdots + f_n

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

f

and

g

are both differentiable, then

Product Rule (Derivatives)

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

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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)

inside the difference quotient for

f(x)g(x)

, splitting it into a piece that isolates

g'

and a piece that isolates

f'

. The fact that differentiability implies continuity ensures that

f(x+h) \to f(x)

h \to 0

.

For three factors:

(fgh)' = f'gh + fg'h + fgh'

. The pattern generalizes—each factor is differentiated once while all others remain, and the results are summed.

The product rule as a growing rectangle

Think of f · g as the area of a rectangle with sides f and g. When x moves a little, f grows by Δf and g grows by Δg. The new area adds a strip g · Δf on the right, a strip f · Δg on top, and a small corner Δf · Δg. Dividing by Δx, the strips give f′g and fg′, while the corner, a product of two small changes, shrinks to nothing.

That is why the product rule has two terms and not f′g′: the corner is the only place both factors change.

Quotient Rule

f

and

g

are differentiable and

g(x) \neq 0

, then

Quotient Rule (Derivatives)

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

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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'

, not

fg' - f'g

.

The quotient rule can be derived from the product rule by writing

f/g = f \cdot g^{-1}

and applying the chain rule to

g^{-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

g

is differentiable at

x

and

f

is differentiable at

g(x)

, then the composite function

f \circ g

is differentiable at

x

and

Chain Rule

\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)

and

u = g(x)

Chain Rule

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Chain Rule

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \qquad \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

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The notation suggests fraction cancellation, and while

dy/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)))

: 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(x^2)

e^{3x}

\ln(\cos x)

—cannot be differentiated. Most applications of differentiation involve the chain rule at some level.

Mean Value Theorem

f

is continuous on

[a, b]

and differentiable on

(a, b)

, then there exists at least one point

c \in (a, b)

such that

Mean Value Theorem

f'(c) = \frac{f(b) - f(a)}{b - a} \qquad \text{for some } c \in (a, b)

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The instantaneous rate of change at

c

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))

and

(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]

ensures no gaps at the endpoints. Differentiability on the open interval

(a, b)

ensures the derivative exists at interior points. Functions that violate either condition may fail the conclusion.

Secant over [a, b] and the parallel tangent at c

The dashed line joins the endpoints of the interval and has slope (f(b) − f(a))/(b − a); the solid tangent at the marked c has the same slope, so the two lines are parallel. The theorem promises such a c on any interval where the function is smooth, and for the parabola there is exactly one. Change the interval and watch c move to keep the tangent parallel on the mean value theorem visualizer.

Somewhere the instantaneous rate equals the average rate: that is the whole theorem, and Rolle’s is its level special case.

Rolle's Theorem

Rolle's Theorem is the special case of the Mean Value Theorem where

f(a) = f(b)

.

If

f

is continuous on

[a, b]

, differentiable on

(a, b)

, and

f(a) = f(b)

, then there exists at least one point

c \in (a, b)

with

f'(c) = 0

Rolle's Theorem

\text{If } f(a) = f(b), \text{ then } f'(c) = 0 \text{ for some } c \in (a, b)

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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) = 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)

. It also appears in arguments about the number of roots: between any two roots of

f

, there must be a root of

f'

Rolle’s theorem on a full period of sine

f(x) = sin x on [0, 2π] starts and ends at 0, so the secant through the endpoints is flat. Rolle’s theorem promises at least one point inside where the tangent is flat too, and here there are two: cos c = 0 at c = π/2, the peak, and c = 3π/2, the trough. The curve rises above its starting height and has to turn back down, and then it dips below and has to turn back up. Try other intervals on the mean value theorem visualizer.

Between two roots of f there is always a root of f′: here between 0 and π lies π/2.

L'Hôpital's Rule

\lim_{x \to a} \frac{f(x)}{g(x)}

produces the indeterminate form

\frac{0}{0}

\frac{\infty}{\infty}

, and if

\lim_{x \to a} \frac{f'(x)}{g'(x)}

exists (or equals

\pm\infty

), then

L'Hôpital's Rule

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \text{for indeterminate forms } \tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty}

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The rule replaces a difficult limit of functions with a (potentially simpler) limit of their derivatives. It applies equally when

a

is finite or

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 \cdot \infty

\infty - \infty

0^0

1^\infty

\infty^0

—can be converted to

\frac{0}{0}

\frac{\infty}{\infty}

through algebraic rearrangement, then handled by L'Hôpital's rule. The conversion step varies by form:

0 \cdot \infty

becomes a fraction by moving one factor to the denominator; exponential forms use the identity

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.

Why the derivatives decide 0/0

Take lim sin(2x)/x as x → 0. Both curves pass through the origin, so the quotient starts as 0/0. Close to 0 each curve is almost its tangent line: sin 2x is almost 2x and x is exactly x. The ratio of the heights is then almost the ratio of the slopes, 2/1. At x = 0.3 the heights are 0.565 and 0.3, a ratio of 1.88, and the ratio closes in on 2 as x shrinks.

L’Hôpital’s rule compares slopes: lim f/g = lim f′/g′ = 2 cos 0 / 1 = 2.

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 · x^{n − 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

Derivative Rules FAQ

When can you use L'Hôpital's rule?

Only when the limit is already an indeterminate form of type 0/0 or ∞/∞, and the limit of the derivative ratio exists or is ±∞. Applying it to a determinate limit produces wrong answers. If the derivative ratio yields another indeterminate form you may apply the rule again; if it oscillates or fails to exist, the rule simply gives no information — it does not show the original limit fails.Read more →

Can you use L'Hôpital's rule on 0 × ∞ or ∞ − ∞?

Not directly — the rule accepts only 0/0 and ∞/∞. The other forms have to be rewritten as a quotient first. For 0 · ∞, move one factor into the denominator; for ∞ − ∞, combine the terms into a single fraction. The exponential forms 0⁰, 1^∞ and ∞⁰ go through the identity f(x)^g(x) = e^(g(x)·ln f(x)), which converts them to 0 · ∞.Read more →

What is the difference between the product rule and the quotient rule?

The product rule adds two terms: (fg)′ = f′g + fg′. The quotient rule subtracts, then divides: (f/g)′ = (f′g − fg′)/g². Order matters in that numerator — it is f′g − fg′, never fg′ − f′g, because subtraction is not commutative. The quotient rule follows from the product rule applied to f · g⁻¹, so the two are not independent.Read more →

How do you apply the chain rule to three nested functions?

Work from the outside in, with each layer contributing one factor. For f(g(h(x))) the derivative is f′(g(h(x))) · g′(h(x)) · h′(x): differentiate the outermost function at its argument, multiply by the derivative of the next layer at its argument, and continue until the innermost function is reached. Any depth of nesting follows the same pattern.Read more →