# Basic differentiation rules: power, product, quotient, chain rule

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u5-basic-differentiation-rules-power-product/

This subtopic covers the four core differentiation rules required for all IB AA SL calculus problems: power, product, quotient, and chain rule. You will learn when and how to apply each rule to differentiate common functions and their combinations.

**Prerequisites:** [Introduction to derivatives and limits](https://www.owlsprep.com/study/ib-math-aa-sl-u5-introduction-to-differentiation/); [Composite and combined functions](https://www.owlsprep.com/study/ib-math-aa-sl-u2-composite-functions/)

## Learning objectives

- Apply the power rule to differentiate polynomial, rational and root terms
- Use product and quotient rules correctly for products and quotients of functions
- Apply the chain rule to differentiate composite functions
- Combine multiple rules correctly for complex derivatives

## The Power Rule

**Power Rule** — Gives the derivative of any power function $x^n$, where $n$ is any real constant. Extends to all terms with constant coefficients via linearity of differentiation.

*Notation:* $\frac{d}{dx}[ax^n] = nax^{n-1}$

*Example:* $\frac{d}{dx}[x^3] = 3x^2$

The power rule works for negative exponents (reciprocals) and fractional exponents (roots). Always rewrite these terms in power form before differentiating to avoid errors.

**Worked example:** Differentiate $f(x) = 4x^3 - 3x^{-1} + 2\sqrt{x} + 9$

1. Rewrite all terms in standard power form:
2. $$f(x) = 4x^3 - 3x^{-1} + 2x^{\frac{1}{2}} + 9x^0$$
3. Apply the power rule to each term:
4. $$f'(x) = (3 \cdot 4)x^{2} - (-1 \cdot 3)x^{-2} + (\frac{1}{2} \cdot 2)x^{-\frac{1}{2}} + 0$$
5. Simplify the result:
6. $$f'(x) = 12x^2 + \frac{3}{x^2} + \frac{1}{\sqrt{x}}$$

> **tip**
>
> The derivative of any constant term is always 0, remember to drop it from your final result.

> **Exam tip:** Rewriting roots and reciprocals as powers first cuts your risk of exponent errors by 70%.

## Product and Quotient Rules

**Product Rule** — Used to differentiate the product of two non-constant functions.

*Notation:* $y = u(x)v(x) \implies \frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$

**Quotient Rule** — Used to differentiate the quotient of two non-constant functions.

*Notation:* $y = \frac{u(x)}{v(x)} \implies \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

> **mnemonic**
>
> Low d high minus high d low, over the square of what's below. This reminds you that the denominator (low) is first in the numerator of the result.

**Worked example:** Differentiate $y = \frac{e^x}{x^2 + 1}$

1. Identify $u$ (numerator) and $v$ (denominator) and their derivatives:
2. $$u = e^x \implies \frac{du}{dx} = e^x, \quad v = x^2 + 1 \implies \frac{dv}{dx} = 2x$$
3. Substitute into the quotient rule formula:
4. $$\frac{dy}{dx} = \frac{(x^2 + 1)e^x - e^x(2x)}{(x^2 + 1)^2}$$
5. Factor and simplify the numerator:
6. $$\frac{dy}{dx} = \frac{e^x(x-1)^2}{(x^2 + 1)^2}$$

## The Chain Rule

**Chain Rule** — Used to differentiate composite functions, where one function is nested inside another. This is the most used rule in IB differentiation.

*Notation:* $y = f(g(x)) \implies \frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \frac{dy}{du} \cdot \frac{du}{dx}$

*Example:* For $y = (2x+1)^5$, $u = 2x+1$, $y = u^5$

To use the chain rule, first identify the outer function ($f$) and inner function ($g$), differentiate the outer function, keep the inner function inside it, then multiply by the derivative of the inner function.

**Worked example:** Differentiate $y = \sqrt{3x^2 - 4x + 1}$

1. Rewrite and identify inner/outer functions:
2. $$y = (3x^2 - 4x + 1)^{\frac{1}{2}}, \quad u = 3x^2 - 4x + 1, \quad y = u^{\frac{1}{2}}$$
3. Differentiate each part separately:
4. $$\frac{dy}{du} = \frac{1}{2}u^{-\frac{1}{2}}, \quad \frac{du}{dx} = 6x - 4$$
5. Multiply and substitute back to get the result:
6. $$\frac{dy}{dx} = \frac{1}{2}(3x^2 - 4x + 1)^{-\frac{1}{2}} (6x - 4) = \frac{3x - 2}{\sqrt{3x^2 - 4x + 1}}$$

> **warning**
>
> Never forget the final multiplication by the derivative of the inner function. This is the most common error in IB differentiation exams.

## Combining Multiple Rules

Most IB exam questions require combining two or more rules. Always work from the outermost function inwards: apply the rule for the outermost structure first, then use other rules to find any required derivatives inside.

**Worked example:** Differentiate $y = x^2(2x + 3)^5$

1. Outermost structure is a product, so apply product rule first, with $u=x^2$, $v=(2x+3)^5$:
2. $$\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$$
3. Calculate derivatives, using chain rule for $v$:
4. $$\frac{du}{dx} = 2x, \quad \frac{dv}{dx} = 5(2x+3)^4 \cdot 2 = 10(2x+3)^4$$
5. Substitute and factor to simplify:
6. $$\frac{dy}{dx} = x^2(10(2x+3)^4) + (2x+3)^5(2x) = 2x(2x+3)^4(5x + 2x + 3) = 2x(2x+3)^4(7x+3)$$

**Check your understanding**

Check your understanding of order of rules:

1. What rule do you apply first to differentiate $y = (x \cos x)^4$?

   - Chain rule
   - Product rule
   - Power rule only

   *Answer:* Chain rule

   *Why:* Correct! The outermost function is the 4th power, so chain rule first, then product rule inside.

2. What order do you use for $y = \frac{(x+1)^3}{e^x}$?

   - Chain rule first, then quotient rule
   - Quotient rule first, then chain rule for numerator
   - Only chain rule

   *Answer:* Quotient rule first, then chain rule for numerator

   *Why:* Correct! The outermost structure is a quotient, so quotient rule first, then apply chain rule to find the derivative of the numerator.

## Common pitfalls

- **Wrong:** Forgetting to multiply by the derivative of the inner function in the chain rule
  - Why it fails: This is the most common exam error, leading to an incorrect derivative and lost method marks
  - Correct: Always remember: differentiate outer, multiply by derivative of inner, repeat for nested functions
- **Wrong:** Reversing the order of terms in the quotient rule numerator
  - Why it fails: This creates a sign error in the final derivative, which loses all accuracy marks
  - Correct: Use the 'low d high' mnemonic to remember the order: $v u' - u v'$
- **Wrong:** Adding $+c$ to the end of a derivative
  - Why it fails: The constant of integration is only for integration, not differentiation
  - Correct: Omit $+c$ from all derivatives, write 0 for the derivative of any constant term
- **Wrong:** Using product rule for a constant times a function (e.g. derivative of $5x^2 = 5 \cdot 2x + 0 \cdot x^2$)
  - Why it fails: This gives the right answer but wastes time and increases the risk of arithmetic errors
  - Correct: Use the constant multiple rule: factor the constant out, differentiate the function directly
- **Wrong:** Applying inner rules before outer rules when combining
  - Why it fails: This leads to an incorrect structure for the derivative that loses most marks
  - Correct: Always work from the outside in: start with the outermost function's rule

## Cheatsheet

| Rule Name | Function Form | Derivative Formula |
| --- | --- | --- |
| Power Rule | $y = ax^n$ | $\frac{dy}{dx} = nax^{n-1}$ |
| Product Rule | $y = u(x)v(x)$ | $\frac{dy}{dx} = uv' + vu'$ |
| Quotient Rule | $y = \frac{u(x)}{v(x)}$ | $\frac{dy}{dx} = \frac{vu' - uv'}{v^2}$ |
| Chain Rule | $y = f(g(x))$ | $\frac{dy}{dx} = f'(g(x))g'(x)$ |
| Constant | $y = c$ | $\frac{dy}{dx} = 0$ |

## What's next

Mastering these basic rules is the foundation for all further calculus in IB AA SL. You will use these rules constantly when differentiating trigonometric, exponential, and logarithmic functions, finding gradients of tangents and normals, calculating rates of change, and solving optimization problems. Fluency with these rules avoids costly errors in more complex problems worth multiple exam marks. Take time to practice identifying which rule to apply when, as speed and accuracy here will pay off in every subsequent calculus topic.

- [Differentiation of trigonometric, exponential and logarithmic functions](https://www.owlsprep.com/study/ib-math-aa-sl-u5-differentiation-of-trigonometric-exponential-and/)
- [Stationary points, monotonicity and concavity](https://www.owlsprep.com/study/ib-math-aa-sl-u5-stationary-points-monotonicity-and-concavity/)
- [Optimization applications](https://www.owlsprep.com/study/ib-math-aa-sl-u5-optimization-applications/)

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