Differentiation rules and composition

Core Rules & Products

When functions combine, we use three essential rules. The product rule finds the derivative of multiplied functions: $$(u\,v)' = u'\,v + u\,v'$$

The quotient rule handles division: $$\left(\frac{u}{v}\right)' = \frac{u'\,v - u\,v'}{v^{2}}$$

Common pitfall: Never differentiate products or quotients term-by-term. Test $f=g=x$ to prove the fake rules fail instantly.

Chain Rule & Limits

The chain rule differentiates composed functions $y = f(g(x))$: $$\frac{dy}{dx} = f'!\bigl(g(x)\bigr)\,\cdot\,g'(x)$$

Strategy: Always identify the inner function $g(x)$ and outer function $f(u)$ explicitly before applying the rule to avoid missing factors.

Worked example: For $h(x) = e^{3x^{2}}$, outer $f(u)=e^{u}$ and inner $g(x)=3x^{2}$ give $h'(x) = 6x\,e^{3x^{2}}$.

L'Hôpital's rule: When limits yield $\frac{0}{0}$ or $\frac{\infty}{\infty}$, evaluate $$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ provided the right-hand limit exists.