The most common chain rule error is writing the outer derivative at x instead of at the inner function. The formula prevents this:
\[\frac{d}{dx}\big[f(g(x))\big]=f'\big(g(x)\big)\cdot g'(x)\]
The first factor is \(f'\big(g(x)\big)\), not \(f'(x)\). The outer function's derivative is evaluated at whatever the inner function produced.
Worked contrast. For \(y=\ln(x^2)\), the outer function is \(\ln(\square)\) and the inner function is \(x^2\).
- Wrong: \(\dfrac{1}{x}\cdot 2x=2\). Here \(\dfrac{1}{x}\) evaluated the log derivative at x, not at the inner function.
- Right: \(\dfrac{1}{x^2}\cdot 2x=\dfrac{2}{x}\). The inner expression \(x^2\) is copied into the denominator unchanged.
The same idea in placeholder form: if the outer function is \(\ln(\square)\), its derivative is \(\dfrac{1}{\square}\), and the placeholder is filled with the inner function exactly as written — nothing differentiated, nothing simplified.
Practical check. After writing the outer factor, look at the expression sitting inside it. It must be character-for-character the inner function you identified before differentiating. If it has been altered, the factor is wrong.