Product Rule: Finding The Derivative Of Y=(5x^2+3)(4x-3)
Hey guys! Ever stumbled upon a function that looks like two smaller functions multiplied together and wondered how to find its derivative? That's where the product rule comes to the rescue! It's a fundamental concept in calculus, and in this article, we're going to break down how to use it step-by-step. We'll focus on a specific example: finding the derivative of y=(5x^2+3)(4x-3). So, grab your calculators, and let's dive in!
Understanding the Product Rule
Before we jump into our specific problem, let's make sure we're all on the same page about what the product rule actually is. In simple terms, the product rule is a formula that helps us find the derivative of a function that is the product of two other functions. Think of it like this: if you have y = u(x) * v(x), where u(x) and v(x) are both functions of x, then the derivative of y with respect to x (that's dy/dx) can be found using this formula:
dy/dx = u'(x) * v(x) + u(x) * v'(x)
Where:
- u'(x) is the derivative of u(x)
- v'(x) is the derivative of v(x)
Basically, the product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function. It might sound a little complicated at first, but trust me, it's easier than it looks! The key is to identify the two functions being multiplied, find their individual derivatives, and then plug everything into the formula. Remember, this rule is super important in calculus because many functions we encounter are actually combinations of simpler functions multiplied together. Mastering the product rule opens the door to differentiating a wider range of functions, which is crucial for solving various problems in physics, engineering, economics, and many other fields. So, pay close attention, and let's get this down!
Applying the Product Rule to y=(5x^2+3)(4x-3)
Okay, now that we've got the product rule formula fresh in our minds, let's put it into action! Our mission is to find the derivative of the function y = (5x^2 + 3)(4x - 3). The first thing we need to do is identify our u(x) and v(x). Looking at the function, it's pretty clear that we can define:
- u(x) = 5x^2 + 3
- v(x) = 4x - 3
Now comes the fun part: finding the derivatives of these two functions! Let's start with u(x). To find u'(x), we'll use the power rule (which states that the derivative of x^n is nx^(n-1)) and the constant rule (the derivative of a constant is zero):
u'(x) = d/dx (5x^2 + 3) = 10x + 0 = 10x
Easy peasy, right? Now, let's tackle v(x). Again, we'll use the power rule and the constant rule:
v'(x) = d/dx (4x - 3) = 4 - 0 = 4
Alright, we've got all the pieces of the puzzle! We know u(x), v(x), u'(x), and v'(x). Now, we just need to plug them into the product rule formula:
dy/dx = u'(x) * v(x) + u(x) * v'(x)
dy/dx = (10x)(4x - 3) + (5x^2 + 3)(4)
We're almost there! Now, let's simplify this expression to get the final derivative.
Simplifying the Derivative
We've successfully applied the product rule and arrived at the expression: dy/dx = (10x)(4x - 3) + (5x^2 + 3)(4). But, to truly find the derivative, we need to simplify this expression as much as possible. This involves a little bit of algebraic manipulation, but don't worry, it's nothing we can't handle!
First, let's distribute the terms in each part of the expression:
(10x)(4x - 3) = 40x^2 - 30x
(5x^2 + 3)(4) = 20x^2 + 12
Now, we can substitute these expanded expressions back into our equation:
dy/dx = (40x^2 - 30x) + (20x^2 + 12)
Next, we combine like terms. This means adding together the terms with the same power of x:
dy/dx = 40x^2 + 20x^2 - 30x + 12
dy/dx = 60x^2 - 30x + 12
And there you have it! We've simplified the expression, and we've found the derivative of y = (5x^2 + 3)(4x - 3) using the product rule:
dy/dx = 60x^2 - 30x + 12
This is the final answer. You might notice that all the coefficients in this derivative (60, -30, and 12) are divisible by 6. We could factor out a 6 to write the derivative in a slightly more compact form:
dy/dx = 6(10x^2 - 5x + 2)
But either form is perfectly acceptable. The key is that we've correctly applied the product rule and simplified the result.
Common Mistakes and How to Avoid Them
Using the product rule is pretty straightforward once you get the hang of it, but there are a few common mistakes that students often make. Knowing these pitfalls can help you avoid them and ensure you get the correct derivative every time. One of the most frequent errors is forgetting to apply the product rule altogether! When you see a function that's the product of two other functions, your first instinct should be to reach for the product rule. Don't try to distribute terms and then differentiate (unless it simplifies the problem significantly), because that will often lead to the wrong answer. Another common mistake is misidentifying u(x) and v(x). Make sure you clearly define which function is which before you start differentiating. A simple way to do this is to write them down explicitly, like we did in our example: u(x) = 5x^2 + 3 and v(x) = 4x - 3. This helps keep things organized and reduces the chance of errors. And of course, mistakes in differentiation of u(x) or v(x) themselves will lead to an incorrect final answer. Double-check your power rule, constant rule, and any other differentiation rules you use. A small error in finding u'(x) or v'(x) will propagate through the rest of the problem. Finally, don't forget to simplify your answer! While the unsimplified derivative is technically correct, it's always good practice to simplify as much as possible. This makes the answer easier to work with and reduces the risk of errors in future calculations.
Practice Makes Perfect
The best way to master the product rule (or any calculus concept, for that matter) is through practice! The more problems you work through, the more comfortable and confident you'll become. So, don't just read through this article and think you've got it. Grab some practice problems from your textbook, online resources, or even make up your own! Try differentiating functions like:
- y = (x^3 + 2x)(x^2 - 1)
- y = (sin x)(cos x)
- y = (ex)(x2 + 1)
For each problem, follow the steps we outlined above: identify u(x) and v(x), find u'(x) and v'(x), apply the product rule formula, and simplify your answer. Don't be afraid to make mistakes! Everyone makes mistakes when they're learning something new. The important thing is to learn from your mistakes and keep practicing. If you get stuck on a problem, go back and review the steps, check your differentiation rules, and make sure you haven't made any algebraic errors. And if you're still stuck, don't hesitate to ask for help from your teacher, a tutor, or a classmate. With enough practice, you'll be a product rule pro in no time!
Conclusion
So, there you have it! We've successfully navigated the product rule and found the derivative of y = (5x^2 + 3)(4x - 3). Remember, the product rule is a powerful tool for differentiating functions that are the product of two other functions. By following the steps we've outlined – identifying u(x) and v(x), finding their derivatives, applying the formula, and simplifying – you can confidently tackle any product rule problem that comes your way. And remember, practice is key! The more you use the product rule, the more natural it will become. So, keep practicing, keep learning, and you'll be a calculus whiz in no time! Keep an eye out for more calculus tips and tricks in future articles. Happy differentiating, guys!