MathematicsCalculusA-Level

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Derivative (power)

Differentiate x^n using the power rule.

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\frac{d}{d x} (x^{n}) = n x^{n - 1}

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Core idea

Overview

The power rule is a foundational principle in calculus used to compute the derivative of a variable raised to a constant real-number exponent. It establishes that the slope of a power function is determined by multiplying the variable term by its current exponent and then decrementing that exponent by exactly one.

When to use: Apply this rule when differentiating any term in the form xⁿ, where n is a constant value. It is valid for all real numbers, including positive integers, negative integers, and fractional exponents representing roots.

Why it matters: This rule allows for the rapid calculation of rates of change without relying on the tedious limit definition of derivatives. It is essential in physics for deriving acceleration from velocity and in economics for determining marginal costs and revenue.

Symbols

Variables

n = Power n, x = Variable x, \frac{dy}{dx} = Derivative value

n

Power n

V a r iab l e

x

Variable x

V a r iab l e

\frac{d y}{d x}

Derivative value

V a r iab l e

Walkthrough

Derivation

Derivation of the Power Rule for Differentiation

The power rule states that the derivative of $x^{n}$ is n x^(n-1). It can be derived from first principles using the binomial expansion.

n is a positive integer for this derivation (so the binomial theorem gives a finite expansion).
The limit as h approaches 0 exists.

Start with First Principles:

Use the definition of the derivative as a limit of a difference quotient.

f^{'} (x) = h \to 0 lim \frac{( x + h ) ^{n} - x ^{n}}{h}

Expand (x+h)^n Using the Binomial Theorem:

Expand the expression into terms with increasing powers of h.

(x + h)^{n} = x^{n} + n x^{n - 1} h + \frac{n ( n - 1 )}{2} x^{n - 2} h^{2} + \dots + h^{n}

Cancel x^n and Divide by h:

Subtracting $x^{n}$ cancels the first term, leaving only terms containing h.

f^{'} (x) = h \to 0 lim \frac{n x ^{n - 1} h + \frac{n ( n - 1 )}{2} x ^{n - 2} h ^{2} + \dots + h ^{n}}{h}

Take the Limit:

As $h \to 0$ , all terms still containing h vanish, leaving only the first term.

f^{'} (x) = h \to 0 lim (n x^{n - 1} + \frac{n ( n - 1 )}{2} x^{n - 2} h + \dots + h^{n - 1})

Final Result:

So $\frac{d}{d x} (x^{n}) = n x^{n - 1}$ .

f^{'} (x) = n x^{n - 1}

Result

f^{'} (x) = n x^{n - 1}

Source: AQA A-Level Mathematics — Pure (Differentiation)

Visual intuition

Graph

Graph unavailable for this formula.

The graph appears as a horizontal line because the derivative dy is equal to the constant power n. Since n remains fixed as you move along the x-axis, the output stays constant regardless of the input value. For a student, this means the rate of change for the power rule is independent of the variable x, showing that the derivative is determined solely by the exponent itself. The most important feature is that the line is perfectly flat, which signifies that the derivative does not scale or shift as the input chang

Graph type: constant

Why it behaves this way

Intuition

The derivative nx^(n-1) describes the slope of the tangent line to the curve y=xn at any given point x, illustrating how the steepness of the curve changes across its domain.

\frac{d}{d x}

The instantaneous rate of change of a function with respect to the variable x.

It tells you how quickly the function's value changes for a tiny change in x, representing the steepness of the function's graph at a specific point.

x^{n}

A power function, where x is the independent variable and n is a constant real-number exponent.

Represents a curve whose steepness and curvature depend on the values of n and x. Common examples include parabolas (x2) or cubics (x3).

The constant exponent to which the variable x is raised in the original function.

It dictates the 'order' or 'degree' of the power function and significantly influences its shape and growth rate.

n x^{n - 1}

The derivative of xn, which gives the slope of the tangent line to the curve y=xn at any point x.

This new function quantifies the exact steepness of the original curve at every point along its path.

Signs and relationships

n-1 (as the exponent in the derivative): The exponent decreases by one because differentiation calculates the rate of change, which is typically one order or 'dimension' lower than the original function. For instance, the rate of change of an area (x2)
n (as the coefficient in the derivative): The original exponent 'n' becomes a multiplicative factor, scaling the rate of change. This reflects how the magnitude of the original exponent directly influences the steepness of the derivative.

Free study cues

Insight

Canonical usage

This rule dictates how the dimension of a power function changes when differentiated with respect to its base variable.

Common confusion

A common mistake is forgetting that if 'x' has units, the derivative 'd/dx( $x^{n}$ )' will have a different unit (specifically, the unit of 'x' raised to 'n-1'), rather than being dimensionless or having the same unit as

Dimension note

If the variable 'x' is dimensionless (e.g., a pure number, a ratio), then ' $x^{n}$ ' is also dimensionless, and its derivative 'nx^(n-1)' will remain dimensionless.

Unit systems

$x$ [U] · If the variable 'x' has a specific unit [U] (e.g., meters, seconds), then 'x^n' will have the unit [U]^n. Consequently, the derivative 'd/dx(x^n)' will have the unit [U]^(n-1).

One free problem

Practice Problem

Calculate the instantaneous rate of change of the function f(x) = x³ at the point where x = 2.

Power n3

Variable x2

Solve for: $d y$

Hint: Apply the power rule nxⁿ⁻¹ by substituting 3 for n and 2 for x.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Finding velocity from displacement equation.

Study smarter

Tips

Multiply the term by the current exponent before reducing the power.
Subtract exactly one from the exponent, ensuring careful calculation with negative numbers.
Convert radical signs into fractional exponents before applying the rule.
Remember that the derivative of a linear term x¹ is simply 1.

Avoid these traps

Common Mistakes

Integrating instead of differentiating.
Forgetting n=0 for constants.

Common questions

Frequently Asked Questions

The power rule states that the derivative of x^n is n x^(n-1). It can be derived from first principles using the binomial expansion.

Apply this rule when differentiating any term in the form xⁿ, where n is a constant value. It is valid for all real numbers, including positive integers, negative integers, and fractional exponents representing roots.

This rule allows for the rapid calculation of rates of change without relying on the tedious limit definition of derivatives. It is essential in physics for deriving acceleration from velocity and in economics for determining marginal costs and revenue.

Integrating instead of differentiating. Forgetting n=0 for constants.