Derivative Of F(x) = 4√(6x² + 3): A Step-by-Step Guide
Hey guys! Today, we're diving into a fun calculus problem: finding the derivative of the function f(x) = 4√(6x² + 3). If you're scratching your head already, don't worry! We'll break it down step by step so it's super clear and easy to follow. Understanding derivatives is crucial in calculus, and this example is a great way to practice applying the chain rule. So, let's grab our pencils and paper and get started!
Understanding Derivatives and the Chain Rule
Before we jump into the specifics of our function, let's quickly recap what derivatives are and why the chain rule is our best friend in this situation. In the most basic terms, a derivative tells us the instantaneous rate of change of a function. Think of it as the slope of a curve at a single point. This concept has massive applications in physics, engineering, economics, and pretty much any field that deals with change and rates.
Now, what about the chain rule? Well, the chain rule is what we use when we need to find the derivative of a composite function – that is, a function within a function. Our function, f(x) = 4√(6x² + 3), is a perfect example of this. We have the square root function acting on the expression 6x² + 3. The chain rule essentially tells us to take the derivative of the "outer" function, keeping the "inner" function as is, and then multiply by the derivative of the "inner" function. It might sound a bit complicated, but once we apply it, you'll see how smoothly it works. It’s like peeling an onion, layer by layer, to get to the core!
Remember, the chain rule is expressed mathematically as: d/dx [f(g(x))] = f'(g(x)) * g'(x). This formula might look intimidating, but it's simply a way of formalizing the process we just described. The first part, f'(g(x)), means we take the derivative of the outer function f, keeping the inner function g(x) unchanged. Then, we multiply by g'(x), which is the derivative of the inner function.
The chain rule is incredibly important because composite functions are everywhere in calculus and beyond. From trigonometric functions nested inside polynomials to exponential functions composed with rational functions, the chain rule allows us to tackle a huge variety of derivative problems. Without it, we'd be stuck trying to find derivatives of only the simplest functions. It’s the Swiss Army knife of differential calculus, a tool you’ll use again and again.
Mastering the chain rule requires practice, so working through examples like our f(x) = 4√(6x² + 3) function is a great way to build your skills. Each time you apply the chain rule, you're reinforcing your understanding of how it works and becoming more confident in your ability to handle more complex problems. So, let's dive into the specifics of our function and see the chain rule in action!
Step-by-Step Solution
Okay, let's get down to business and find the derivative of f(x) = 4√(6x² + 3). We'll break it down into manageable steps so you can follow along easily. Trust me; it's not as scary as it looks!
Step 1: Rewrite the function
First things first, let's rewrite the square root using exponents. This will make it easier to apply the power rule later on. Remember that the square root of something is the same as raising it to the power of 1/2. So, we can rewrite our function as:
f(x) = 4(6x² + 3)^(1/2)
This simple change makes the function look less intimidating and sets us up perfectly for using the chain rule. It's a little trick that makes a big difference in how we approach the problem. Guys, rewriting functions in a more convenient form is a common strategy in calculus, and it's something you'll get better at with practice. It's like learning to speak the language of calculus fluently!
Step 2: Apply the Chain Rule
Now comes the fun part – applying the chain rule! As we discussed earlier, the chain rule tells us to differentiate the outer function first, keeping the inner function the same, and then multiply by the derivative of the inner function. In our case, the "outer" function is 4u^(1/2) (where u represents the inner function 6x² + 3), and the "inner" function is 6x² + 3.
Let's start by differentiating the outer function. Using the power rule, which states that the derivative of x^n is nx^(n-1), we get:
d/du [4u^(1/2)] = 4 * (1/2) * u^((1/2) - 1) = 2u^(-1/2)
Notice that we've kept the inner function u as it is for now. We've simply applied the power rule to the outer function. This is a crucial step in the chain rule process. Remember, we're peeling the onion layer by layer, so we focus on the outer layer first.
Next, we need to find the derivative of the inner function, 6x² + 3. This is a straightforward application of the power rule and the constant rule (the derivative of a constant is zero):
d/dx [6x² + 3] = 12x
We've now found the derivatives of both the outer and inner functions. The last piece of the puzzle is to put them together using the chain rule formula:
f'(x) = [derivative of outer function] * [derivative of inner function]
So, we have:
f'(x) = 2u^(-1/2) * 12x
Step 3: Substitute and Simplify
We're almost there! The last thing we need to do is substitute 6x² + 3 back in for u and simplify the expression. Remember, u was just a placeholder to make the chain rule easier to apply.
Substituting u = 6x² + 3, we get:
f'(x) = 2(6x² + 3)^(-1/2) * 12x
Now, let's simplify this. First, we can multiply the constants 2 and 12x together:
f'(x) = 24x(6x² + 3)^(-1/2)
To make the expression look cleaner, we can rewrite the term with the negative exponent as a fraction. Remember that x^(-n) = 1/x^n. So, we have:
f'(x) = 24x / (6x² + 3)^(1/2)
Finally, we can rewrite the term in the denominator as a square root again:
f'(x) = 24x / √(6x² + 3)
And there you have it! We've found the derivative of f(x) = 4√(6x² + 3). Give yourself a pat on the back – you've successfully navigated the chain rule!
Step 4: Further Simplification (Optional but Recommended)
While we've found the derivative, we can often simplify it further to make it even cleaner and easier to work with. In this case, we can factor out a 3 from inside the square root in the denominator:
f'(x) = 24x / √(3(2x² + 1))
Now, we can use the property of square roots that √(ab) = √a * √b to separate the square root:
f'(x) = 24x / (√3 * √(2x² + 1))
We can also simplify the fraction by dividing both the numerator and denominator by a common factor. Notice that 24 in the numerator and √3 in the denominator share a common factor related to √3. To see this clearly, we can rewrite 24 as 8 * 3, and then rewrite 3 as √3 * √3:
f'(x) = 8 * 3 * x / (√3 * √(2x² + 1)) = 8 * √3 * √3 * x / (√3 * √(2x² + 1))
Now we can cancel out one of the √3 terms:
f'(x) = 8√3 * x / √(2x² + 1)
This is a simplified form of the derivative, and it's often easier to work with in further calculations. So, simplifying your derivatives whenever possible is a good habit to develop.
Common Mistakes to Avoid
When working with derivatives and the chain rule, there are a few common pitfalls that students often stumble into. Being aware of these mistakes can help you avoid them and ensure you get the correct answer. Let's take a look at some of the most frequent errors:
Forgetting the Chain Rule
This is probably the most common mistake. When you have a composite function, you must use the chain rule. Forgetting to multiply by the derivative of the inner function will lead to an incorrect result. Always double-check if you have a function within a function, and if you do, remember the chain rule!
Incorrectly Applying the Power Rule
The power rule is a fundamental tool in differentiation, but it's easy to make a mistake if you're not careful. Remember that the power rule states that the derivative of x^n is nx^(n-1). Make sure you correctly multiply by the exponent and subtract 1 from it. A common error is forgetting to subtract 1 from the exponent, especially when dealing with fractional or negative exponents.
Messing Up the Order of Operations
The chain rule involves multiple steps, so it's crucial to follow the correct order of operations. Differentiate the outer function first, keeping the inner function the same, and then multiply by the derivative of the inner function. Mixing up the order can lead to a completely wrong answer.
Not Simplifying the Result
While finding the derivative is the main goal, simplifying your answer is often necessary, especially if you need to use the derivative in further calculations. Simplification can involve combining like terms, factoring, or rationalizing denominators. Leaving your answer unsimplified can make subsequent steps more difficult and increase the chance of errors.
Sign Errors
Sign errors are easy to make, especially when dealing with negative signs in the exponents or coefficients. Pay close attention to the signs throughout your calculations, and double-check your work to catch any mistakes. A small sign error can completely change your result.
Incorrectly Identifying the Inner and Outer Functions
In order to apply the chain rule effectively, you need to correctly identify the inner and outer functions. Sometimes, this is straightforward, but in more complex functions, it can be tricky. Take your time to carefully analyze the function and determine which part is the "outer" function and which is the "inner" function.
By being mindful of these common mistakes, you can improve your accuracy and confidence in finding derivatives using the chain rule. Remember, practice makes perfect, so the more you work through problems, the better you'll become at avoiding these pitfalls.
Practice Problems
Want to really nail down your understanding of the chain rule? The best way to do that is by working through practice problems! Here are a few for you to try. Work them out yourself, and then you can check your answers with a friend or your instructor. Remember, the key is to apply the chain rule step by step, and don't be afraid to make mistakes – that's how we learn!
- g(x) = (3x² - 2x + 1)^5
- h(x) = sin(4x³)
- k(x) = e^(x² + 1)
- m(x) = √(5x - 2)
- n(x) = ln(x² + 3x)
These problems cover a range of functions, including polynomials, trigonometric functions, exponential functions, and logarithmic functions. Working through them will give you a solid understanding of how to apply the chain rule in different situations. Don't just rush through the problems – take your time to understand each step and why you're doing it. The goal is not just to get the right answer, but to truly understand the process.
If you get stuck on a problem, don't get discouraged! Go back and review the steps we covered in the example, or look up the chain rule in your textbook or online. There are tons of resources available to help you learn. And remember, the more you practice, the easier it will become. Calculus is like learning a new language – it takes time and effort, but it's totally worth it in the end!
Conclusion
So, guys, we've successfully navigated the world of derivatives and the chain rule! We tackled the problem of finding the derivative of f(x) = 4√(6x² + 3) step by step, and along the way, we reinforced our understanding of some key calculus concepts. Remember, the chain rule is your friend when it comes to composite functions, and breaking down complex problems into smaller, manageable steps is always a winning strategy.
Calculus can seem intimidating at first, but with practice and a solid understanding of the fundamental rules, you can conquer any derivative problem that comes your way. Keep practicing, stay curious, and don't be afraid to ask questions. You've got this!
Happy differentiating!