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Derivatives

The Product Rule

4 min readnovember 23, 2021

nicholeajacquez

Nichole Jacquez


Calculus 🤓

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Using the Product Rule

The product rule is crucial in deriving complex equations and word problems, and it’s a rule that you use all the time in your calculus classes! By understanding the product rule, you’ll become better equipped to handle multiplication problems. 😌

Understanding Derivatives ✅

To understand the product rule, we must first know what derivatives are! 🧠
After all, the product rule is a way of finding the derivative of multiplying functions. If you don’t need a refresher, feel free to skip to the product rule section. 
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Derivatives are helpful tools in finding what the slope is at a certain point. This is also known as the tangent at a point (remember, the tangent is Y/X, or rise/run)! 🌄
For example, the image above displays the slope of the blue dot. It's crucial to recognize that this isn't the slope of the entire function but just the slope at the blue point. Finding slopes at specific points can help mathematicians (like yourself) find out just how much a function changes over a specific interval! ⛄

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👊 Refresher on the Power Rule 

The product rule requires you to know how to find the derivatives of functions, so let’s review how to do that! 😀
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The power rule is what you will often think about when deriving a function. Let’s use the function x^4 as an example. Using the image as a reference, n = 4, so you’ll multiply that in the front, which makes 4x. You're not done, though! You have to take the original n (4) and subtract 1 from it. This, then, makes the solution 4x^3! In other words, the derivative (dy/dx or d/dx) of x^4 = 4x^3. 🎉

😲 What if there are NO exponents?

When a function has no exponent attached to it, that is the same as the function being to the  1st power (or power of 1). For example, x is the same as x^1. When you find the derivative of that, the n would be 0 after subtracting 1. Anything to the power of 0 is 1. That means that the derivative of any term without an exponent will be 1 through the power rule. 1️⃣

🔗 What if a function in parentheses is raised to a power?

You would use the chain rule, which you can find details of here!
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Product Rule ❌

❔ Which functions can I apply the Product Rule to? 

The product rule is exactly what its name implies: it applies to equations that use products, also known as multiplication problems! 😳
Examples of multiplication problems: 
  • 3x * 5x^2
  • 3x^2 * 4x^3
  • 5x * 6x^3
Going deeper, the product rule goes like this:
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💡 Note: “DRight” and “DLeft” mean that those are the derivatives of the left and right terms.
Let’s apply this rule to an example multiplication problem from earlier!
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💡 Memory Tip: What works as a mnemonic (a pattern to help recall information) is saying “Left dee-right plus right dee-left”! 🎵
Still stumped? Check out another approach to the product rule here. ⚡

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🙅‍♂️ Common Mistakes with the Product Rule

  • Multiplying the left term with its own derivative and vice versa (i.e Left-DLeft + Right-DRight). Make sure that you put the left term with the derivative of the right term, and the right term with the derivative of the left!
  • Writing terms to the 0 power like 5^0 as the original term (in this case, 5) or as 0. Remember, anything to the 0th power is 1! 5^0 is 1.
  • Writing terms to the 1 power like 3^1 as 1. Anything to the 1st power is just the number or term itself, so 3^1 is 3! 

🔨 Tips for Mastering the Product Rule

The number one (and only one you need to know) tip you need to take away from this post: Remember your exponent rules! 😉
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Image courtesy of Wikipedia

  1. Multiplying exponents means that you add them: x^2 * x^3 = x^(2+3) = x^5. ➕
  2. Dividing exponents means that you subtract them: x^7 / x^4 = x^(7-4) = x^3. ➖
  3. Multiplying an exponent within parentheses by an exponent means that you multiply them: (x^2)^5 = x^(2 * 5) = x^10. ❌

📝 Practice Problems: Apply Your Knowledge!

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🔑 Answer Key with Detailed Steps:

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Expanding Your Knowledge 📖

If the product rule hasn’t come easy—don’t sweat it! Practice makes progress, and even just tackling the practice problems has progressed your derivative skills and math skills in general! Tackle this video on the product and quotient rule for some extra practice. 💯
Once you’ve gotten comfortable with the product rule, try tackling other rules like quotient rule and chain rule! As you progress through calculus concepts, you will have to use multiple of these rules in tandem with each other—so get acquainted when you can! 🤝
Happy deriving! 😊
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