Differentiation rules and composition

When functions are combined, we differentiate using three core rules.

Product rule

$$(u\,v)' = u'\,v + u\,v'$$

Quotient rule

$$\left(\frac{u}{v}\right)' = \frac{u'\,v - u\,v'}{v^{2}}$$

Chain rule — for composed functions $y = f(g(x))$:

$$\frac{dy}{dx} = f'\!\bigl(g(x)\bigr)\;\cdot\;g'(x)$$

Worked example

Differentiate $h(x) = e^{3x^{2}}$.

• Outer: $f(u)=e^{u}$, so $f'(u)=e^{u}$.
• Inner: $g(x)=3x^{2}$, so $g'(x)=6x$.
• Result: $h'(x) = 6x\,e^{3x^{2}}$.

L'Hôpital's rule — When a limit gives $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$:

$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$

provided the right-hand limit exists.

Strategy: Always identify the inner and outer functions before applying the chain rule. Write them out explicitly to avoid sign and factor errors.

Common pitfall: $(fg)' \ne f'g'$ and $(f/g)' \ne f'/g'$. Test any tempting rule on a trivial example like $f = g = x$ before trusting it — the fake product rule fails instantly.