Derivative Of (x+3)^4/√(x^2+5): Step-by-Step Solution

by Rajiv Sharma 54 views

Hey guys! Today, we're diving into a fun calculus problem: finding the derivative of the function f(x) = (x+3)^4 / √(x^2+5). This might look a bit intimidating at first glance, but don't worry! We'll break it down step by step using the quotient rule and the chain rule. So, grab your pencils and let's get started!

1. Understanding the Function and the Rules

Before we jump into the calculations, let's make sure we're all on the same page. Our function is a fraction where both the numerator and the denominator are functions of x. This means we'll definitely need the quotient rule. Remember the quotient rule? It states that if you have a function h(x) = f(x) / g(x), then its derivative h'(x) is given by:h'(x) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2. Also, we'll see the chain rule pop up when we're dealing with the composite functions (functions within functions), such as (x+3)^4 and √(x^2+5). The chain rule tells us that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x).

Breaking Down the Function

Let's identify our f(x) and g(x) in this case:

  • f(x) = (x+3)^4 (the numerator)
  • g(x) = √(x^2+5) (the denominator)

Now we need to find the derivatives of f(x) and g(x) separately. This is where the chain rule comes into play. To find the derivative of f(x) = (x+3)^4, we first recognize that we have an outer function (something raised to the power of 4) and an inner function (x+3). Applying the chain rule:

  • The derivative of the outer function (u^4) is 4u^3.
  • The derivative of the inner function (x+3) is 1.

So, f'(x) = 4(x+3)^3 * 1 = 4(x+3)^3. Remember, the chain rule is super useful when you have a function inside another function. Think of it like peeling an onion – you deal with the outer layer first, then move inwards.

Now, let's tackle g(x) = √(x^2+5). We can rewrite this as g(x) = (x2+5)(1/2). Again, we'll use the chain rule:

  • The derivative of the outer function (u^(1/2)) is (1/2)u^(-1/2).
  • The derivative of the inner function (x^2+5) is 2x.

So, g'(x) = (1/2)(x2+5)(-1/2) * 2x = x / √(x^2+5). It's crucial to practice these basic derivative rules until they become second nature. Knowing the power rule, chain rule, and quotient rule like the back of your hand will make more complex problems much easier to handle. And remember, math is like building blocks – you need a strong foundation to construct something impressive.

Why These Rules Matter

The quotient rule and chain rule aren't just abstract formulas; they're essential tools for understanding how functions change. In real-world applications, derivatives help us analyze rates of change, optimize processes, and model complex systems. For example, in physics, derivatives are used to calculate velocity and acceleration. In economics, they can help determine marginal cost and revenue. By mastering these fundamental concepts, you're not just learning math; you're gaining valuable skills that can be applied in many different fields.

2. Applying the Quotient Rule

Okay, we've got our f(x), g(x), f'(x), and g'(x). Now it's time to plug them into the quotient rule formula: h'(x) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2. Let’s substitute what we found earlier:

h'(x) = [4(x+3)^3 * √(x^2+5) - (x+3)^4 * (x / √(x^2+5))] / [√(x2+5)]2

This looks a bit messy, right? Don't panic! We're going to simplify this expression step by step. Simplifying complex expressions is a key skill in calculus. It not only makes the result easier to understand but also reduces the chance of making errors in further calculations. So, let’s roll up our sleeves and get to it!

Step-by-Step Simplification

First, let's deal with the denominator. [√(x2+5)]2 simply becomes (x^2+5). That was easy! Now, let’s focus on the numerator. We have two terms separated by a minus sign. To combine them, we need a common denominator. Notice that the second term has √(x^2+5) in its denominator, but the first term doesn't. So, we'll multiply the first term by √(x^2+5) / √(x^2+5):

h'(x) = [4(x+3)^3 * √(x^2+5) * (√(x^2+5) / √(x^2+5)) - (x+3)^4 * (x / √(x^2+5))] / (x^2+5)

Now, we can simplify the first term: √(x^2+5) * √(x^2+5) becomes (x^2+5). So, the numerator now looks like this:

4(x+3)^3 * (x^2+5) / √(x^2+5) - (x+3)^4 * (x / √(x^2+5))

Now both terms in the numerator have the same denominator, √(x^2+5). We can combine them:

[4(x+3)^3 * (x^2+5) - (x+3)^4 * x] / √(x^2+5)

Remember, when simplifying complex fractions, it's often helpful to break the problem down into smaller, more manageable steps. Look for common factors, combine like terms, and don't be afraid to rewrite the expression in different ways. Practice makes perfect, so the more you simplify, the better you'll get at it!

Combining Fractions and Simplifying

Now, we have a single fraction in the numerator and a denominator outside. To simplify further, we can think of dividing by (x^2+5) as multiplying by 1/(x^2+5). So, we can rewrite our expression as:

h'(x) = {[4(x+3)^3 * (x^2+5) - (x+3)^4 * x] / √(x^2+5)} * [1 / (x^2+5)]

This combines into a single fraction:

h'(x) = [4(x+3)^3 * (x^2+5) - (x+3)^4 * x] / [√(x^2+5) * (x^2+5)]

See how much cleaner it's starting to look? We're not done yet, but we're making progress. The key here is to take your time, double-check each step, and look for opportunities to simplify. Math is like a puzzle – each step brings you closer to the solution.

3. Simplifying the Expression

Alright, we've got our derivative in a somewhat simplified form, but we can still do more! Let's focus on the numerator: 4(x+3)^3 * (x^2+5) - (x+3)^4 * x. Notice that both terms have a common factor of (x+3)^3. Let's factor that out:

h'(x) = [(x+3)^3 * [4(x^2+5) - (x+3)x]] / [√(x^2+5) * (x^2+5)]

Factoring is a powerful technique in algebra and calculus. It helps us simplify expressions, solve equations, and identify common structures. By factoring out (x+3)^3, we've made the numerator much easier to handle. Now, let's simplify the expression inside the brackets:

Expanding and Combining Terms

Inside the brackets, we have 4(x^2+5) - (x+3)x. Let's expand these terms:

4(x^2+5) = 4x^2 + 20 -(x+3)x = -x^2 - 3x

Now, combine them:

4x^2 + 20 - x^2 - 3x = 3x^2 - 3x + 20

So, our derivative now looks like this:

h'(x) = [(x+3)^3 * (3x^2 - 3x + 20)] / [√(x^2+5) * (x^2+5)]

We're getting closer to the final form! The numerator is now fully simplified. Let’s turn our attention to the denominator. Remember that (x^2+5) is the same as (x2+5)1. We also have √(x^2+5), which is the same as (x2+5)(1/2). When multiplying terms with the same base, we add the exponents:

Dealing with the Denominator

So, √(x^2+5) * (x^2+5) = (x2+5)(1/2) * (x2+5)1 = (x2+5)(1/2 + 1) = (x2+5)(3/2)

Now, our derivative looks like this:

h'(x) = [(x+3)^3 * (3x^2 - 3x + 20)] / (x2+5)(3/2)

And that’s it! We've successfully simplified the derivative. This final form is much cleaner and easier to work with than our initial expression. Remember, simplification is not just about getting the right answer; it's about understanding the structure of the expression and making it more manageable.

4. The Final Answer

So, after all that hard work, here's our final answer for the derivative of f(x) = (x+3)^4 / √(x^2+5):

f'(x) = [(x+3)^3 * (3x^2 - 3x + 20)] / (x2+5)(3/2)

Reviewing the Steps

Let's quickly recap the steps we took to get here:

  1. Identified f(x) and g(x) and recognized the need for the quotient rule.
  2. Found the derivatives f'(x) and g'(x) using the chain rule.
  3. Applied the quotient rule formula.
  4. Simplified the expression step by step, factoring, expanding, and combining terms.
  5. Arrived at the final simplified form of the derivative.

This problem is a great example of how calculus often involves a combination of different rules and techniques. It also highlights the importance of careful simplification. Don't be afraid to take your time, break the problem down into smaller parts, and double-check your work along the way.

Practice Makes Perfect

If you found this a bit challenging, don't worry! Calculus takes practice. Try working through similar problems, and gradually increase the complexity. The more you practice, the more comfortable you'll become with these concepts. And remember, there are tons of resources available online and in textbooks to help you along the way. Don't hesitate to seek out help when you need it. Learning calculus is a journey, not a race. Enjoy the process, and celebrate your successes!

Conclusion

Finding the derivative of (x+3)^4 / √(x^2+5) might have seemed daunting at first, but by breaking it down into manageable steps and applying the quotient and chain rules, we were able to conquer it! Remember, calculus is all about understanding the fundamental rules and applying them strategically. Keep practicing, and you'll be a derivative master in no time! You've got this, guys!