Deriving the General Power Rule

Why a Power of a Function Needs More Than the Power Rule

A familiar differentiation rule says that the derivative of \(x^n\) is \(nx^{n-1}\). When the base is itself a function, such as \((x^2+1)^3\), there are two changes to account for: the outer power changes, and the inner function changes as \(x\) changes.

The Chain Rule combines those changes. The goal here is to derive the resulting formula and make its conditions explicit, so it can be applied without losing the contribution from the inner function.

Key idea: Differentiate the outer power as though its input were a single variable, then multiply by the derivative of that input.

The General Power Rule

Let \(n\) be a fixed real number, and let \(u\) be differentiable on an open interval \(I\). If \(u(x)>0\) for every \(x\in I\), then the real power \(u(x)^n\) is defined throughout \(I\), and the general power rule is

\frac{d}{dx}[u(x)]^n = n[u(x)]^{n-1}u'(x).

The positivity condition is a straightforward domain condition that works for every real exponent \(n\). For example, a non-integer real power of a negative number is not generally a real-valued expression. For integer exponents, the formula also applies in some cases where \(u(x)\) is not positive:

If \(n\) is a positive integer, the formula applies wherever \(u\) is differentiable, including where \(u(x)=0\).
If \(n\) is a negative integer, the formula applies wherever \(u(x)\ne 0\), because the power is not defined at zero.
If \(n=0\), the expression is the constant \(1\) wherever it is defined, so its derivative is \(0\).

These integer cases extend the formula beyond positive inputs; they do not remove the need for the expression to be defined. In the examples below, each domain will be specified or will follow from the choice of exponent and inner function.

Deriving the Formula from the Chain Rule

Define an outer function \(p(t)=t^n\). For positive \(t\), the ordinary power rule gives \(p'(t)=nt^{n-1}\). Now write the composite expression as \(p(u(x))\). Since \(u\) is differentiable and its values are positive on \(I\), both the outer derivative and the Chain Rule apply there.

\begin{aligned} \frac{d}{dx}[u(x)]^n &= \frac{d}{dx}\bigl(p(u(x))\bigr)\\ &= p'(u(x))u'(x)\\ &= n[u(x)]^{n-1}u'(x). \end{aligned}

This proves the formula under the stated real-exponent conditions. The key substitution is \(t=u(x)\) in the outer derivative \(p'(t)=nt^{n-1}\). The Chain Rule then supplies the additional factor \(u'(x)\).

Why the inner derivative appears: A change in \(x\) first changes \(u(x)\), and that change affects the outer power. The factor \(n[u(x)]^{n-1}\) measures the outer function’s rate of change at the current input; \(u'(x)\) measures how quickly that input changes.

Worked Examples

Worked Example: A Power of a Polynomial

Differentiate \(f(x)=(x^2+1)^3\), and check the result by expanding first.

Set \(u(x)=x^2+1\) and \(n=3\). Then \(u'(x)=2x\). The inner function is positive for every real \(x\), so the rule applies:

\begin{aligned} f'(x) &=3(x^2+1)^{3-1}(2x)\\ &=6x(x^2+1)^2. \end{aligned}

For an independent check, expand the original polynomial and differentiate its terms:

\begin{aligned} (x^2+1)^3 &=(x^2+1)(x^2+1)(x^2+1)\\ &=x^6+3x^4+3x^2+1,\\ \frac{d}{dx}(x^6+3x^4+3x^2+1) &=6x^5+12x^3+6x. \end{aligned}

Factoring this derivative gives

$$ 6x^5+12x^3+6x =6x(x^4+2x^2+1) =6x(x^2+1)^2. $$

This matches the result from the general power rule.

Worked Example: A Negative Integer Exponent

Differentiate \(g(x)=(2x+3)^{-2}\). The expression is defined when \(2x+3\ne0\), or \(x\ne-\frac{3}{2}\). Take \(u(x)=2x+3\), so \(u'(x)=2\), and take \(n=-2\):

\begin{aligned} g'(x) &=-2(2x+3)^{-3}(2)\\ &=-4(2x+3)^{-3}. \end{aligned}

Check the value of the derivative at \(x=0\) directly from the difference quotient. Here \(g(0)=3^{-2}=\frac{1}{9}\):

\begin{aligned} \frac{g(h)-g(0)}{h} &=\frac{(2h+3)^{-2}-\frac{1}{9}}{h}\\ &=\frac{\frac{1}{(2h+3)^2}-\frac{1}{9}}{h}\\ &=\frac{9-(2h+3)^2}{9(2h+3)^2h}\\ &=\frac{9-(4h^2+12h+9)}{9(2h+3)^2h}\\ &=\frac{-4h^2-12h}{9(2h+3)^2h}\\ &=\frac{-4(h+3)}{9(2h+3)^2}. \end{aligned}

Taking the limit as \(h\to0\) gives \(-4(3)/(9\cdot 9)=-\frac{4}{27}\). The formula gives \(g'(0)=-4(3)^{-3}=-\frac{4}{27}\), as required.

Worked Example: A Non-Integer Exponent

Differentiate \(q(x)=(x^2+4)^{1/2}\). Since \(x^2+4\ge4>0\), the inner function is positive for every real \(x\), so the real-exponent version applies. With \(u(x)=x^2+4\), \(u'(x)=2x\), and \(n=\frac12\), we obtain

\begin{aligned} q'(x) &=\frac12(x^2+4)^{\frac12-1}(2x)\\ &=x(x^2+4)^{-\frac12}\\ &=\frac{x}{\sqrt{x^2+4}}. \end{aligned}

To check at \(x=0\), the formula gives \(q'(0)=0/\sqrt{4}=0\). The difference quotient at zero confirms this:

\begin{aligned} \frac{q(h)-q(0)}{h} &=\frac{\sqrt{h^2+4}-2}{h}\\ &=\frac{(\sqrt{h^2+4}-2)(\sqrt{h^2+4}+2)} {h(\sqrt{h^2+4}+2)}\\ &=\frac{h^2}{h(\sqrt{h^2+4}+2)}\\ &=\frac{h}{\sqrt{h^2+4}+2}. \end{aligned}

As \(h\to0\), the numerator tends to \(0\) and the denominator tends to \(4\), so this quotient tends to \(0\). It agrees with the derivative formula at \(x=0\).

A Common Mistake: Omitting the Inner Derivative

A frequent error is to differentiate the outer power but stop before multiplying by the derivative of the inner function. For example, an incorrect calculation is

\frac{d}{dx}(x^2+1)^4 \overset{\text{wrong}}{=}4(x^2+1)^3.

This treats \(x^2+1\) as though it changed at a rate of \(1\). In fact, its derivative is \(2x\). The correct calculation is

\begin{aligned} \frac{d}{dx}(x^2+1)^4 &=4(x^2+1)^3\frac{d}{dx}(x^2+1)\\ &=4(x^2+1)^3(2x)\\ &=8x(x^2+1)^3. \end{aligned}

A quick numerical check exposes the omission at \(x=1\). The incorrect expression gives \(4(2)^3=32\), while the correct expression gives \(8(1)(2)^3=64\). The inner derivative factor is essential.

Using the Rule Reliably

For an expression that is a power of a function, identify the exponent and the entire base before differentiating. Then differentiate that base separately. The following sequence keeps the roles distinct:

Identify the inner function.
Write the base as \(u(x)\), including all of its terms or factors.

Identify the exponent.
Write the fixed exponent as \(n\), and confirm that the power is defined on the interval being considered.

Differentiate the outer power.
Write \(n[u(x)]^{n-1}\).

Multiply by the inner derivative.
Attach \(u'(x)\), then simplify if useful.

The exponent decreases by one, while the derivative of the inner function is multiplied as a separate factor. These are two different operations: changing the exponent does not differentiate the base.

Why the Rule Is Useful

Many expressions encountered in differentiation are powers of nontrivial functions: a polynomial inside a power, a positive expression raised to a fractional power, or a nonzero expression raised to a negative integer power. The general power rule handles all of these using the same structure. It also makes the Chain Rule visible: the outer derivative alone is not enough, and the inner derivative records how the input to the outer function changes.

Before applying the rule, check its domain conditions. For an arbitrary real exponent, a positive inner function guarantees that the real power is defined and differentiable. For positive integer exponents, zero and negative inner values are allowed. For negative integer exponents, zero must be excluded. Stating these conditions prevents a correct-looking differentiation step from being used where the original expression is not defined.

Check Your Understanding

For each question, identify the inner function and check that the power is defined before differentiating.

Differentiate \((3x^2-1)^5\), showing the inner derivative as a separate factor.
On which real \(x\)-values is \((x-4)^{-3}\) defined, and what is its derivative there?
Explain why \((x^2+2)^{1/3}\) satisfies the positive-inner-function condition for a real exponent.
What factor is missing from the incorrect expression \(\frac{d}{dx}(5x+1)^6=6(5x+1)^5\)?
For \(u(x)>0\), explain how the Chain Rule changes the outer derivative \(nt^{n-1}\) into the derivative of \([u(x)]^n\).

Deriving the General Power Rule

What You'll Learn

Why a Power of a Function Needs More Than the Power Rule

The General Power Rule

Deriving the Formula from the Chain Rule