# Differentiation rules and composition

Physics I · Calculus of a Single Variable · https://tryals.app/learn/physics-i/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:
$$\left(\frac{u}{v}\right)' = \frac{u'\,v - u\,v'}{v^{2}}$$

| Operation | Incorrect Trap | Correct Formula | 
| --- | --- | --- | 
| **Product** | $(fg)' \neq f'g'$ | $u'v + uv'$ | 
| **Quotient** | $(f/g)' \neq \frac{f'}{g'}$ | $\frac{u'v - uv'}{v^2}$ | 

**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.

## Practice questions

4 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. The chain rule multiplies derivatives of the outer and inner functions. What does that multiplication encode about the way rates combine?

A. That the two derivatives are independent and so may be combined freely
B. That rates compose, and the middle units cancel
C. That composition is the same operation as multiplication for functions
D. That the inner function must be linear for the rule to apply

**Answer:** B. That rates compose, and the middle units cancel

**Why:** Chained rates multiply, and the intermediate unit cancels. This is why $dz/dx = (dz/dy)(dy/dx)$ reads correctly as a sentence about how fast things change relative to each other.

Page: https://tryals.app/practice/physics-i/differentiation-rules-and-composition/the-chain-rule-multiplies-derivatives-of-the-outer-and-inner

### 2. Match each differentiation rule to its formula.

**Answer:**

- Power rule → $d/dx[x^{n}] = n\cdot x^{n-1}$
- Chain rule → $d/dx[f(g(x))]$ = f'($g(x)$)$\cdot g$'(x)
- Quotient rule → $d/dx[f/g]$ = (f'g − fg')$/g^{2}$
- Product rule → $d/dx[f\cdot g]$ = f'g + fg'

**Why:** These four rules cover the differentiation of most elementary expressions.

Page: https://tryals.app/practice/physics-i/differentiation-rules-and-composition/match-each-differentiation-rule-to-its-formula

### 3. Which rules are needed to differentiate $h(x) = x^{2}\cdot \cos(3x)$?

A. Quotient rule
B. Product rule
C. Chain rule
D. Power rule

**Answer:** B. Product rule; C. Chain rule; D. Power rule

**Why:** Product rule for $x^{2}$·$\cos(3x)$; chain rule for $\cos(3x)$ (inner derivative 3); power rule for $x^{2}$.

Page: https://tryals.app/practice/physics-i/differentiation-rules-and-composition/which-rules-are-needed-to-differentiate-h-x-x-cos-3x

### 4. Fill in the key terms about differentiation rules.

**Answer:** The **product rule** gives d/dx[fg] = f'g + fg'. The **quotient rule** gives $d/dx[f/g]$ = (f'g − fg')$/g^{2}$. For composite functions the **chain rule** multiplies the outer and inner derivatives.

**Why:** Each rule has a distinct structural pattern for recognising when to apply it.

Page: https://tryals.app/practice/physics-i/differentiation-rules-and-composition/fill-in-the-key-terms-about-differentiation-rules