Power Rule for Derivatives

The power rule is one of the very first shortcuts you learn in differentiation, and it is also one of the most used tools in all of calculus. It tells you how to find the derivative of any function written as \(x\) raised to a fixed exponent, without going back to the formal limit definition of a derivative every single time.

In symbols, the power rule says that if \(f(x) = x^n\), where \(n\) is any real number, then:

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

In words: bring the exponent down to the front as a multiplier, then subtract 1 from the exponent. That is the entire rule. Once you can do this quickly, differentiating polynomials, roots, and reciprocal powers becomes a matter of seconds rather than minutes.

You do not need to memorize the power rule blindly. It comes directly from the limit definition of the derivative, \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\). For a simple case like \(f(x) = x^2\), expanding \((x+h)^2\) gives \(x^2 + 2xh + h^2\), so:

\(f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} (2x + h) = 2x\)

Notice that \(2x\) matches exactly what the power rule predicts: bring the exponent 2 down, then reduce the exponent by 1 to get \(x^1\). The same pattern holds for any whole number exponent, and it can be extended using other differentiation techniques to negative and fractional exponents as well. That general proof is beyond what you need for everyday use, but it is worth knowing that the rule is not just a trick, it is a proven consequence of the limit definition covered in lessons like slope and equation of tangent line.

Example 1: Differentiate \(f(x) = x^5\).

Bring the exponent down and subtract 1: \(f'(x) = 5x^{5-1} = 5x^4\).

Example 2: Differentiate \(f(x) = 7x^3\).

The constant 7 is a multiplier and stays put. Apply the power rule to \(x^3\) and multiply: \(f'(x) = 7 \cdot 3x^{3-1} = 21x^2\).

Two small rules travel alongside the power rule and let you differentiate whole polynomials term by term.

Constant multiple rule: if \(f(x) = a \cdot x^n\), then \(f'(x) = a \cdot n \cdot x^{n-1}\). The constant \(a\) just rides along.

Derivative of a constant: if \(f(x) = c\), a plain number with no \(x\), then \(f'(x) = 0\). A constant function never changes, so its rate of change, its slope, is always zero.

Example 3: Differentiate \(f(x) = 4x^3 - 2x^2 + 9x - 6\).

Differentiate each term separately: \(f'(x) = 12x^2 - 4x + 9 - 0 = 12x^2 - 4x + 9\).

The power rule applies just as well when the exponent is negative, which covers functions written with \(x\) in the denominator.

Example 4: Differentiate \(f(x) = \frac{1}{x^2}\).

Rewrite using a negative exponent: \(f(x) = x^{-2}\). Apply the rule: \(f'(x) = -2x^{-2-1} = -2x^{-3} = -\frac{2}{x^3}\).

The key habit here is to rewrite any denominator power of \(x\) as a negative exponent before differentiating, since the power rule only reads exponents that are already sitting on \(x\).

Square roots, cube roots, and other radicals can be rewritten as fractional exponents, which brings them into the power rule as well.

Example 5: Differentiate \(f(x) = \sqrt{x}\).

Rewrite as \(f(x) = x^{1/2}\). Apply the rule: \(f'(x) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}\).

A few slip ups show up again and again with the power rule:

Forgetting to subtract 1 from the exponent after bringing it down, leaving the exponent unchanged by mistake.

Mixing up the power rule with the rule for exponential functions like \(2^x\), where the base is fixed and the exponent is the variable. Those need the separate approach shown in derivative of exponential functions, not the power rule.

Applying the power rule directly to a composite expression such as \((3x+1)^4\) without adjusting for the inner function. That situation calls for the chain rule, discussed in the lesson on the chain rule.

Because the power rule always produces another power of \(x\), you can apply it again to the result. Differentiating \(f'(x)\) gives the second derivative \(f''(x)\), and so on. This idea of applying the power rule more than once is developed fully in higher order derivatives, where it is used to study concavity and acceleration.

\(\frac{d}{dx}\left[x^n\right] = n x^{n-1}\) for any real number \(n\).

\(\frac{d}{dx}\left[a x^n\right] = a n x^{n-1}\)

\(\frac{d}{dx}\left[c\right] = 0\) for any constant \(c\).

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