# Derivatives of Product Functions

## Definition of Product Functions

**Definition:** A *product function* consists of two functions multiplied together.

**Examples:**
* **$f(x) = x \sin(x)$** is the product of the function $a(x) = x$ and $b(x) = \sin(x)$. **We would say it as "x times $\sin(x)$"**
* $q(x) = x^5 3^x$ is the product of the function $a(x) = x^5$ and $b(x) = 3^x$
* $p(x) = x^4(x^2 + 1)$ is the product of $a(x) = x^4$ and $b(x) = x^2 + 1$

## Product Rule Formula

**Formula for Product Rule (Prime Notation):**
If $f(x) = a(x) \cdot b(x)$, then $f'(x) = a'(x)b(x) + a(x)b'(x)$.

**Formula for Product Rule (using [Leibnitz Notation](https://youtu.be/noaqytzrRZs):)** If $y = a \cdot b$, then $\displaystyle\frac{dy}{dx} = \displaystyle\frac{da}{dx}b + a\displaystyle\frac{db}{dx}$.

:warning: The derivative rule for the product of functions is probably not what you would think. This is different than sum rule. The derivative of a product is **not** the product of the derivatives.


For an explanation of the rule, you can watch this video:
<iframe width="560" height="315" src="https://www.youtube.com/embed/plcy4PGt6vI" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

## Example Problems

### Example 1: If $f(x) = 3^x \sin(x)$, what is $f'(x)$?

<iframe width="560" height="315" src="https://www.youtube.com/embed/k3jmSJRd4mA" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

### Example 2: If $f(x) =$ : 🦆 $(x) \cdot$ :clown_face: $(x)$, what is $f'(x)$?

**Answer:** :duck:prime times :clown_face: + :duck: times :clown_face:prime

**

### Example 3(a): If $f(x) = x^5 (x^2 + 1)$, what is $f'(x)$?

### Example 3(b): If $f(x) = x^5 (x^2 + 1)$, what is $f'(x)$?

### Example 4: If $f(x) = 3^x 4^x$, what is $f'(x)$?

### Example 5: If $f(x) = x^2 \sin(x)$, what is $f'(x$?


**Solution:** We let $A = x^2$ and $B = \sin(x)$, so $A' = 2x$ and $B' = \cos(x)$. So we have $A'B + AB' = 2x \sin(x) + x^2 \cos(x)$.

**Answer:** $f'(x) = 2x \sin(x) + x^2 \cos(x)$.

**Does order matter?** Not really here. There are several equivalent ways of writing these answers. For instance, we could also have written $f'(x) = \sin(x) 2x + \cos(x) x^2$,and it would still be correct.

### Example 6: If $f(x) = \left( \sin(x) \right)^2$, what is $f'(x)$?


**Solution:** To use Product Rule here, we need to rewrite $\left( \sin(x) \right)^2$ as $\sin(x) \sin(x)$. Then we have
$$ A = \sin(x) \text{ and } B = \sin(x)$$ with $$A' = \cos(x) \text{ and } B' = \cos(x)$$.

So $A'B + AB' = \cos(x) \sin(x) + \sin(x) \cos(x)$, which we can rewrite as $2 \sin(x) \cos(x)$.

**Answer:** $f'(x) = 2 \sin(x) \cos(x)$.

## :warning: *Be careful!* There is a difference between product of functions and function composition.

### * <span style = "color:red"> **Composition, not Multiplication:** </span> $R(x) = \sin(e^x)$ is "sine of (e to the x)". There is no multiplication in play here; instead $e^x$ is being plugged into the function $\sin(x)$.