Product Rule for Products of Functions

TL;DR

When you need to differentiate a function that's made up of two simpler functions multiplied together, the Product Rule is your go-to tool. It helps you break down the problem by taking turns differentiating each part while keeping the other part untouched. This rule saves you from having to expand complex products before differentiating, making the process much more efficient.

1. The Mental Model

Imagine you have two friends, A and B, working together. When you need to check their combined effort, you first check friend A's work while friend B relaxes, and then you check friend B's work while friend A relaxes. You then combine their individual contributions.

2. The Core Material

The Product Rule is essential when you encounter a function, let's call it h(x), that is the product of two other functions, say f(x) and g(x). So, h(x) = f(x) * g(x).

The rule states that the derivative of h(x), written as h'(x), is:

h'(x) = f'(x) * g(x) + f(x) * g'(x)

Let's break down what this means:

• f'(x) is the derivative of the first function, f(x).
• g(x) is the original second function.
• f(x) is the original first function.
• g'(x) is the derivative of the second function, g(x).

Essentially, you differentiate the first function and multiply it by the original second function. Then, you add that to the original first function multiplied by the derivative of the second function.

Steps for Applying the Product Rule

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1. Identify f(x) and g(x): Clearly separate the two functions being multiplied.
2. Find f'(x): Differentiate f(x).
3. Find g'(x): Differentiate g(x).
4. Assemble the parts: Plug f(x), g(x), f'(x), and g'(x) into the Product Rule formula.

graph TD
    A["Identify h(x) = f(x) * g(x)"] --> B["Identify f(x)"]
    A --> C["Identify g(x)"]
    B --> D["Calculate f'(x)"]
    C --> E["Calculate g'(x)"]
    D & E --> F["Apply Formula: f'(x)g(x) + f(x)g'(x)"]
    F --> G["Simplify Result"]

3. Worked Example

Let's find the derivative of h(x) = (3x² + 5) * (e^x).

1. Identify f(x) and g(x):

   • f(x) = 3x² + 5
   • g(x) = e^x

2. Find f'(x):

   • The derivative of 3x² is 2 * 3x^(2-1) = 6x.
   • The derivative of 5 (a constant) is 0.
   • So, f'(x) = 6x + 0 = 6x.

3. Find g'(x):

   • The derivative of e^x is simply e^x.
   • So, g'(x) = e^x.

4. Assemble the parts using the formula f'(x) * g(x) + f(x) * g'(x):

   • h'(x) = (6x) * (e^x) + (3x² + 5) * (e^x)

5. Simplify the result:

   • Notice that e^x is common to both terms. You can factor it out:
   • h'(x) = e^x (6x + 3x² + 5)
   • Often, it's good practice to write the polynomial in descending order of powers:
   • h'(x) = e^x (3x² + 6x + 5)

4. Key Takeaways

• The Product Rule helps you differentiate functions that are products of two other functions.
• The formula is h'(x) = f'(x) * g(x) + f(x) * g'(x).
• Always identify f(x) and g(x) first, then find their individual derivatives.
• The order of the terms in the sum doesn't matter (because addition is commutative), but following the f'g + fg' pattern can help avoid errors.
• Don't forget to simplify your final answer by factoring if possible.
• This rule is different from the Chain Rule, which applies to composite functions (functions within functions).

Common mistakes to avoid:
- Differentiating both functions and multiplying them: This is incorrect. It's not f'(x) * g'(x).
- Not identifying f(x) and g(x) correctly: Make sure you clearly split the original function into two distinct parts.
- Forgetting to include the original functions: Remember it's f'(x) * **g(x)** and **f(x)** * g'(x). You use the original f and g in the formula.
- Algebra errors during simplification: Pay close attention to factoring and combining like terms.

5. Now Try It

Take 15 minutes to find the derivative of y = (4x - 1) * cos(x).
Think about what f(x) and g(x) are, then find f'(x) and g'(x), and finally assemble them into the Product Rule formula.
Success looks like correctly applying the Product Rule and presenting the derivative in its simplest form.