Why the Inner Derivative Matters
A common differentiation error is to apply the power rule to a whole expression inside parentheses and stop too soon. For example, the power rule suggests part of the derivative of \((3x^2+1)^5\), but it does not account for how quickly the inside expression changes with \(x\).
The chain rule supplies that missing information. For a composite power, the derivative of the inner function must multiply the derivative of the outer power. Checking for this factor is a quick way to catch a frequent mistake.
The Chain Rule for a Power of a Function
A composite function is formed by applying one function to the output of another. In \([u(x)]^n\), the inner function is \(u(x)\), and the outer function is \(v\mapsto v^n\). The power rule differentiates the outer function; the chain rule then accounts for the inner function.
The first two factors, \(n[u(x)]^{n-1}\), come from differentiating the outer power. The remaining factor, \(u'(x)\), is the derivative of the inner function. If the inside is simply \(x\), then \(u'(x)=1\), so the factor does not change the expression. That familiar case can make it easy to overlook the factor when the inside is more complicated.
Worked Example: Differentiating \((3x^2+1)^5\)
Worked Example: Differentiating \((3x^2+1)^5\)
Find the derivative and check that the derivative of the inner function is included.
Step 1: Identify the inner function. Let \(u(x)=3x^2+1\). Then \(u'(x)=6x\), since \(\frac{d}{dx}(3x^2)=6x\) and \(\frac{d}{dx}(1)=0\).
Step 2: Differentiate the outer power. The outer function raises its input to the fifth power, so its derivative contributes \(5[u(x)]^4\).
Step 3: Multiply by the inner derivative.
Check: The outer-power contribution is \(5(3x^2+1)^4\), and the inner derivative is \(6x\). Their product is \(30x(3x^2+1)^4\), as found. In particular, the answer includes the factor \(6x\); differentiating only the outside would miss it.
Why the Factor Must Be There
The chain rule states that if \(f\) is differentiable at \(u(x)\) and \(u\) is differentiable at \(x\), then
For a power, take \(f(v)=v^n\). The power rule gives \(f'(v)=nv^{n-1}\). Substitution into the chain rule gives \(n[u(x)]^{n-1}u'(x)\). Thus the factor is not an optional adjustment: it is part of the chain rule.
Here is a direct derivation for positive integer powers. It shows explicitly where the inner derivative enters. Assume \(u\) is differentiable at \(x\), and let \(n\geq 1\). By the difference-of-powers identity,
As \(h\) approaches \(0\), the first factor approaches \(u'(x)\), by the definition of the derivative of \(u\). Differentiability implies continuity, so \(u(x+h)\) approaches \(u(x)\). Each of the \(n\) terms in the sum therefore approaches \([u(x)]^{n-1}\). The sum approaches \(n[u(x)]^{n-1}\). Taking the limit of the product gives
This derivation shows why the inner derivative appears: the difference quotient for the composite contains the difference quotient for \(u\). Its limit is \(u'(x)\), which remains as a factor in the derivative.
Worked Example: A Linear Inner Function
Worked Example: A Linear Inner Function
Differentiate \(g(x)=(2x-3)^4\).
Step 1: Identify and differentiate the inside. Set \(u(x)=2x-3\). Then \(u'(x)=2\), because the derivative of \(2x\) is \(2\) and the derivative of \(-3\) is \(0\).
Step 2: Differentiate the fourth power and multiply by \(u'(x)\).
Check: Applying the rule to the stated inside gives \(4(2x-3)^3\) from the outer power and \(2\) from the inner derivative. Multiplying \(4\cdot 2=8\) confirms the result. The inner factor is present even though it has been combined with the numerical coefficient.
Worked Example: An Inner Derivative That Varies with \(x\)
Worked Example: An Inner Derivative That Varies with \(x\)
Differentiate \(p(x)=(x^2-4x+7)^3\).
Step 1: Differentiate the inner polynomial. Let \(u(x)=x^2-4x+7\). Term by term, \(u'(x)=2x-4+0=2x-4\).
Step 2: Apply the composite-power rule.
Check: Differentiating the outer cube contributes \(3(x^2-4x+7)^2\); differentiating the inside contributes \(2x-4=2(x-2)\). Multiplication gives \(6(x^2-4x+7)^2(x-2)\), matching the answer. The \(x\)-dependent factor cannot be replaced by a constant.
A Common Mistake and Its Correction
A tempting but incorrect calculation for the first example is
This calculation differentiates the outer power and then stops. It treats the inside as though it did not change with \(x\). But \(3x^2+1\) is not constant: its derivative is \(6x\). Applying the chain rule gives
A useful correction is to label both contributions before simplifying: write “outer derivative” and “inner derivative,” then multiply them. This makes it harder to stop after differentiating only the outside.
The check is about the structure of the derivative, not whether a factor is visually written in a particular place. For instance, \(5(3x^2+1)^4(6x)\) and \(30x(3x^2+1)^4\) are the same expression. A correct derivative may have its inner factor combined with a constant or rearranged. What matters is that the factor has been accounted for.
A Reliable Factor Check
When checking a derivative of a composite power, use the following sequence. It focuses on the specific risk of losing the inner derivative.
Write \(u(x)\) for the expression being raised to the power.
For a positive integer power \(n\), this contributes \(n[u(x)]^{n-1}\).
Compute \(u'(x)\) separately, so it is not confused with the outer derivative.
Form \(n[u(x)]^{n-1}u'(x)\), then combine factors only after both contributions are present.
This check applies directly to composite powers. If the inner derivative equals \(1\), the resulting factor does not change the expression. If it equals a constant such as \(2\), it may combine with the coefficient from the outer derivative. If it depends on \(x\), it typically remains as an \(x\)-dependent factor, though algebraic simplification may change how it appears.
The method also helps distinguish a missing factor from a factor that happens to be zero at a particular input. For example, in the first worked example the inner derivative \(6x\) equals zero at \(x=0\), but it is still part of the derivative formula for all \(x\). Checking the formula symbolically avoids deciding that a factor is unnecessary just because it vanishes at one point.
Why This Check Is Useful
The power rule is often first learned with expressions such as \(x^5\). In that case the inner function is \(u(x)=x\), and its derivative is \(1\). The chain-rule factor is present in the rule, but multiplying by \(1\) leaves no visible trace. With a composite expression, that factor becomes important: the inside may change at a rate different from \(1\).
A reliable final question is: What is the derivative of the expression inside the power, and where is it in my answer? If the answer is absent, recheck the chain rule. If it has been combined into a coefficient or another factor, the expression may still be correct; verify by writing the product before simplifying.
Check Your Understanding
For each question, focus on whether the inner derivative has been included.
- Identify the inner function and its derivative in \((5x^2-1)^4\).
- Differentiate \((5x^2-1)^4\), showing the outer derivative and inner derivative before simplifying.
- A student writes \(\frac{d}{dx}(x^3+2)^2=2(x^3+2)\). What factor is missing, and what is the corrected derivative?
- Differentiate \((4x+1)^3\). Explain why a numerical factor from the inner derivative appears in the final coefficient.
- Why does the derivative of \(x^5\) not visibly include an extra factor from the inside?
