Calculus I - Week 3

Derivative Rules: Product, Quotient, and Chain

This week equips students with the three essential rules for differentiating composite and combined functions. The product rule, quotient rule, and chain rule together allow differentiation of nearly any function built from elementary pieces.

Learning Objectives

- State and apply the product rule. - State and apply the quotient rule. - Recognize a composite function and identify its inner and outer parts. - Apply the chain rule to composite functions. - Combine rules when multiple rules are needed in one problem. - Differentiate functions of the form (f(x))^n using the chain rule.

1. Why We Need More Rules

The power rule differentiates x^n easily. But what about x^2 * sin(x), or (x^2 + 1)^10, or (x^3 + 1)/(x^2 - 1)? These functions are built by multiplying, dividing, or composing simpler functions. The power rule alone cannot handle them. We need the product rule, quotient rule, and chain rule.

2. The Product Rule

If h(x) = f(x) * g(x), then: h'(x) = f'(x) * g(x) + f(x) * g'(x) Memory device: 'first times derivative of second, plus second times derivative of first.' Example: Differentiate h(x) = x^2 * (3x + 1). Let f(x) = x^2 and g(x) = 3x + 1. f'(x) = 2x, g'(x) = 3. h'(x) = 2x(3x + 1) + x^2(3) = 6x^2 + 2x + 3x^2 = 9x^2 + 2x.

3. Another Product Rule Example

Differentiate h(x) = (x^3 + 2x)(x^2 - 5). f(x) = x^3 + 2x, f'(x) = 3x^2 + 2. g(x) = x^2 - 5, g'(x) = 2x. h'(x) = (3x^2 + 2)(x^2 - 5) + (x^3 + 2x)(2x) Expand: = 3x^4 - 15x^2 + 2x^2 - 10 + 2x^4 + 4x^2 = 5x^4 - 9x^2 - 10.

4. The Quotient Rule

If h(x) = f(x) / g(x), then: h'(x) = [f'(x) * g(x) - f(x) * g'(x)] / [g(x)]^2 Memory device: 'low d-high minus high d-low, over low-squared.' Example: Differentiate h(x) = x^2 / (x + 1). f(x) = x^2, f'(x) = 2x. g(x) = x + 1, g'(x) = 1. h'(x) = [2x(x + 1) - x^2(1)] / (x + 1)^2 = [2x^2 + 2x - x^2] / (x + 1)^2 = (x^2 + 2x) / (x + 1)^2.

5. Another Quotient Rule Example

Differentiate h(x) = (3x - 1) / (x^2 + 2). f(x) = 3x - 1, f'(x) = 3. g(x) = x^2 + 2, g'(x) = 2x. h'(x) = [3(x^2 + 2) - (3x - 1)(2x)] / (x^2 + 2)^2 = [3x^2 + 6 - 6x^2 + 2x] / (x^2 + 2)^2 = (-3x^2 + 2x + 6) / (x^2 + 2)^2.

6. What Is a Composite Function?

A composite function is a function inside another function. If f(x) = x^10 and g(x) = x^2 + 1, then f(g(x)) = (x^2 + 1)^10. The outer function is f (raise to the 10th power) and the inner function is g (x^2 + 1). Another example: h(x) = sin(x^2). The outer function is sin and the inner is x^2.

7. The Chain Rule

If h(x) = f(g(x)), then: h'(x) = f'(g(x)) * g'(x) In words: the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. Example: h(x) = (x^2 + 1)^5. Outer: f(u) = u^5, f'(u) = 5u^4. Inner: g(x) = x^2 + 1, g'(x) = 2x. h'(x) = 5(x^2 + 1)^4 * 2x = 10x(x^2 + 1)^4.

8. More Chain Rule Examples

Example 1: h(x) = (3x - 2)^7. Outer derivative: 7(3x - 2)^6. Inner derivative: 3. h'(x) = 7(3x - 2)^6 * 3 = 21(3x - 2)^6. Example 2: h(x) = sqrt(x^2 + 9) = (x^2 + 9)^(1/2). Outer derivative: (1/2)(x^2 + 9)^(-1/2). Inner derivative: 2x. h'(x) = (1/2)(x^2 + 9)^(-1/2) * 2x = x / sqrt(x^2 + 9).

9. Combining Rules

Sometimes you need to apply more than one rule. Example: Differentiate h(x) = x^2 * (x^3 + 1)^4. This is a product of x^2 and (x^3 + 1)^4. Use the product rule and chain rule together. Let f = x^2, g = (x^3 + 1)^4. f' = 2x. g' = 4(x^3 + 1)^3 * 3x^2 = 12x^2(x^3 + 1)^3. h'(x) = 2x(x^3 + 1)^4 + x^2 * 12x^2(x^3 + 1)^3 = 2x(x^3 + 1)^4 + 12x^4(x^3 + 1)^3 = 2x(x^3 + 1)^3 [(x^3 + 1) + 6x^3] = 2x(x^3 + 1)^3 (7x^3 + 1).

10. Common Mistakes

- Product rule: forgetting to add both terms — you cannot just multiply derivatives. - Quotient rule: subtracting in the wrong order (numerator must be f'd times g minus f times g'd). - Chain rule: forgetting to multiply by the derivative of the inner function. - Chain rule: applying the outer derivative to x instead of to the inner function. - Trying to use the power rule on (x^2 + 1)^5 without the chain rule — the inner function is not just x.