Why a Correct-Looking Derivative Can Be Wrong
A chain-rule error often leaves the outer part of a derivative looking right. For example, differentiating a sine gives a cosine, and differentiating an exponential \(e^u\) gives \(e^u\). But if the input \(u\) depends on \(x\), its derivative must also appear.
With several nested operations, there may be more than one such factor. A dependable check is to trace the expression inward: at each layer, differentiate the current outer function and multiply by the derivative of its input. If a layer's input changes with \(x\), that layer contributes a factor.
A Layer-by-Layer Check
Suppose a function has nested form \(y=F(G(H(x)))\). The chain rule gives
The factors correspond to the successive layers: the derivative of the outer function, the derivative of the middle function, and the derivative of the innermost function. For a deeper composition, continue in the same way until the input is \(x\).
This factor list is useful for diagnosis. If a proposed derivative has the right outer derivative but is missing \(H'(x)\), for instance, it has accounted for the outer and middle layers but not the innermost one. Do not rely only on whether the answer looks familiar; match each factor to a layer.
Rewrite the function as named layers, starting with the innermost input.
Record the derivative of each layer with respect to its own input.
Use the chain rule to multiply the layer derivatives, with each evaluated at the appropriate input.
Match every factor in the proposed derivative to a layer. If one is absent, restore it and simplify only afterward.
Why Every Inner Factor Is Required
The chain rule follows from how changes pass through successive functions. If \(u=g(x)\) and \(y=f(u)\), a small change in \(x\) changes \(u\), and the change in \(u\) then changes \(y\). The rate of change from \(x\) to \(y\) is the product of the two local rates.
For several layers, apply this relationship at each step. For example, if \(v=h(x)\), \(u=g(v)\), and \(y=f(u)\), then
Each factor is necessary because each describes one link in the dependence of \(y\) on \(x\). Leaving out a factor breaks that chain of rates. This also explains why an inner derivative can be \(1\), or can be \(0\) at a particular input: it still belongs in the formula, even if its value later makes no numerical change or makes the whole derivative zero.
Worked Example: A Polynomial Power with a Missing Factor
Worked Example: A Polynomial Power with a Missing Factor
Differentiate \(f(x)=(3x^2-2)^5\). A proposed answer is \(15(3x^2-2)^4\). Find the missing factor and check the correction.
First identify the layers. Let \(u=3x^2-2\), so \(f=u^5\). The outer derivative with respect to \(u\) is \(5u^4\). The inner derivative is
Therefore the complete derivative is
The proposed answer includes the outer factor \(5(3x^2-2)^4\), but omits the inner factor \(6x\). To check by layers, differentiating \(u^5\) gives \(5u^4\), and passing from \(u\) to \(x\) contributes \(du/dx=6x\); their product is exactly \(30x(3x^2-2)^4\).
As a numerical check at \(x=1\), \(u=3(1)^2-2=1\) and \(du/dx=6(1)=6\). Thus \(f'(1)=5(1)^4\cdot6=30\). The proposed expression gives \(15(1)^4=15\), confirming that it is missing a factor.
Worked Example: A Trigonometric Function of a Composite Angle
Worked Example: A Trigonometric Function of a Composite Angle
Differentiate \(g(x)=\sin(2x^2+3x)\). A proposed answer is \(\cos(2x^2+3x)\). Correct it and check the result.
Set \(u=2x^2+3x\), so \(g=\sin u\). Differentiating the outer function gives \(\cos u\). Differentiating its input gives
Multiplying the two factors gives
The proposed answer has the correct outer derivative, but it leaves out \(du/dx=4x+3\). At \(x=1\), the angle is \(2+3=5\), and the inner derivative is \(4+3=7\), so the complete derivative has value \(7\cos(5)\). The proposed expression gives only \(\cos(5)\). The factor list—\(\cos(u)\) from the sine layer and \(4x+3\) from its input—confirms the correction.
Worked Example: Two Nested Inputs in an Exponential
Worked Example: Two Nested Inputs in an Exponential
Differentiate \(h(x)=e^{(x^2+1)^3}\). A proposed answer is \(3(x^2+1)^2e^{(x^2+1)^3}\). Locate the missing factor and check the full derivative.
There are three layers. Let \(u=x^2+1\), let \(v=u^3\), and let \(h=e^v\). The outer exponential contributes \(e^v\), and the power layer contributes
The innermost layer contributes
Multiplying all three layer derivatives gives
The proposed answer accounts for differentiating \(e^v\) and \(u^3\), but omits \(du/dx=2x\). At \(x=1\), \(u=2\), \(v=8\), and the three factors are \(e^8\), \(3(2)^2=12\), and \(2(1)=2\). Their product is \(24e^8\). The proposed expression gives \(12e^8\). The layer-by-layer product confirms that the corrected derivative includes the missing factor \(2x\).
When a Missing Factor Is a Constant
An inner derivative does not have to contain \(x\). For example, the input \(5x-1\) has derivative \(5\). Therefore, differentiating \(\cos(5x-1)\) gives \(-5\sin(5x-1)\), not merely \(-\sin(5x-1)\). The factor is still required even though it is constant.
Likewise, an inner derivative may equal \(1\), as for the input \(x\). In that case, omitting the factor does not alter the written value, but the chain-rule reasoning still includes it. A useful check is to record the factor before simplifying; this makes it less likely that a nontrivial constant will disappear unnoticed.
A Common Mistake: Stopping at the Outer Derivative
A frequent error is to recognize the familiar outer derivative and stop. For \(y=e^{(4x-1)^2}\), for instance, the incorrect calculation might be
This differentiates the exponential with respect to its exponent and differentiates the square with respect to \(4x-1\), but it stops before differentiating \(4x-1\) with respect to \(x\). The omitted factor is \(4\). Correcting the calculation gives
At \(x=1\), the exponent is \(9\), and the factors are \(e^9\), \(2(3)=6\), and \(4\). Their product is \(24e^9\), while the incorrect expression gives \(6e^9\). The difference comes precisely from the missing derivative of the innermost input.
Why the Factor Check Matters
A missing factor can cause more than a constant-scale error. It may change where a derivative is zero. In the polynomial example, the correct derivative contains \(x\), while the proposed derivative did not. At \(x=0\), the correct derivative is zero because the inner input has derivative \(6x=0\); the proposed expression would not reflect that behavior.
For this reason, avoid checking only whether the answer has the right general appearance. The reliable test is structural: list the nested inputs, write down the derivative contributed by each, and confirm that all of them occur in the product. If one is zero at a particular value, keep it in the formula until after the factors have been accounted for.
Check Your Understanding
For each function, identify the inner derivative factor that must appear. Then write the complete derivative.
- Differentiate \(p(x)=(x^2-4)^6\). Which factor would be missing from \(6(x^2-4)^5\)?
- Differentiate \(q(x)=\cos(3x^2-x)\). What factor accompanies \(-\sin(3x^2-x)\)?
- Differentiate \(r(x)=e^{(2x+1)^3}\). List all layer factors before simplifying.
- A proposed derivative for \(\sin(7x)\) is \(\cos(7x)\). What factor is missing, and what is the corrected derivative?
- Explain why a factor should still be included in the chain-rule calculation when its value is zero at a particular input.
