Rate Of Change: F(x) = (√x - 2) / (√x + 2) At X = 2

Hey guys! Ever stumbled upon a function that looks a bit intimidating and wondered how to figure out its rate of change? Well, today we're diving deep into a fascinating example: f(x) = (√x - 2) / (√x + 2). This function might seem complex at first glance, but don't worry, we're going to break it down step by step. We'll not only find the general rate of change but also pinpoint the rate of change at a specific point, x = 2. So, grab your thinking caps, and let's get started!

Understanding the Rate of Change

Before we jump into the calculations, let's quickly recap what we mean by the "rate of change." In mathematical terms, the rate of change of a function at a particular point is given by its derivative at that point. The derivative, often denoted as f'(x), represents the instantaneous rate at which the function's output changes with respect to its input. Think of it as the slope of the tangent line to the function's graph at that specific point. Knowing the rate of change is super useful in various fields, from physics to economics, helping us understand how things are changing dynamically. For our function f(x), we need to find its derivative, f'(x), and then evaluate it at x = 2 to get the rate of change at that particular point.

Step 1: Finding the Derivative f'(x)

To find the derivative of f(x) = (√x - 2) / (√x + 2), we'll need to employ the quotient rule. This rule is essential when dealing with functions that are expressed as a ratio of two other functions. The quotient rule states that if we have a function h(x) = u(x) / v(x), then its derivative h'(x) is given by: h'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]². In our case, u(x) = √x - 2 and v(x) = √x + 2. So, let's find the derivatives of u(x) and v(x) separately.

Finding u'(x)

Our u(x) is √x - 2. Remember that √x can be written as x^(1/2). To find the derivative u'(x), we'll use the power rule, which states that if g(x) = x^n, then g'(x) = nx^(n-1). Applying the power rule to √x, we get (1/2)x^((1/2)-1) = (1/2)x^(-1/2) = 1 / (2√x). The derivative of the constant -2 is simply 0. Therefore, u'(x) = 1 / (2√x).

Finding v'(x)

Our v(x) is √x + 2. This is very similar to u(x), and we can apply the same logic. The derivative of √x is again 1 / (2√x), and the derivative of the constant 2 is 0. So, v'(x) = 1 / (2√x).

Applying the Quotient Rule

Now that we have u(x), v(x), u'(x), and v'(x), we can plug them into the quotient rule formula: f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]². Substituting the values, we get:

f'(x) = [(1 / (2√x)) * (√x + 2) - (√x - 2) * (1 / (2√x))] / (√x + 2)²

This looks a bit messy, but we can simplify it. Let's first focus on the numerator. Distribute the terms and combine like terms:

Numerator = [(√x + 2) / (2√x) - (√x - 2) / (2√x)]

Since we have a common denominator, we can combine the fractions:

Numerator = [(√x + 2 - (√x - 2)) / (2√x)] = [4 / (2√x)] = 2 / √x

Now, let's put it all together back into the quotient rule formula:

f'(x) = (2 / √x) / (√x + 2)²

To simplify further, we can rewrite this as:

f'(x) = 2 / [√x * (√x + 2)²]

So, we've found the derivative of our function! f'(x) = 2 / [√x * (√x + 2)²]. This expression gives us the rate of change of f(x) at any value of x.

Step 2: Evaluating f'(x) at x = 2

Now that we have the general expression for the rate of change, f'(x), we want to find the specific rate of change when x = 2. This means we simply need to substitute x = 2 into our derivative expression:

f'(2) = 2 / [√2 * (√2 + 2)²]

Let's break this down. First, we have √2 in the denominator. Then, we have (√2 + 2)². To calculate this, we can expand the square:

(√2 + 2)² = (√2)² + 2 * √2 * 2 + 2² = 2 + 4√2 + 4 = 6 + 4√2

So now our expression looks like:

f'(2) = 2 / [√2 * (6 + 4√2)]

We can distribute the √2 in the denominator:

f'(2) = 2 / (6√2 + 4 * 2) = 2 / (6√2 + 8)

To rationalize the denominator, we can multiply both the numerator and the denominator by the conjugate of the denominator, which is 6√2 - 8:

f'(2) = [2 * (6√2 - 8)] / [(6√2 + 8) * (6√2 - 8)]

Let's calculate the new numerator and denominator separately.

New Numerator

2 * (6√2 - 8) = 12√2 - 16

New Denominator

(6√2 + 8) * (6√2 - 8) is in the form (a + b)(a - b), which equals a² - b². So,

(6√2)² - 8² = 36 * 2 - 64 = 72 - 64 = 8

Now we have:

f'(2) = (12√2 - 16) / 8

We can simplify this by dividing both terms in the numerator by 8:

f'(2) = (3√2 - 4) / 2

So, the rate of change of f(x) at x = 2 is (3√2 - 4) / 2. This is the final answer!

Conclusion

Wow, we've done it! We successfully found the rate of change of the function f(x) = (√x - 2) / (√x + 2) and determined its value at x = 2. We started by understanding the concept of rate of change and its connection to derivatives. Then, we used the quotient rule to find the derivative of our function, which was a crucial step. Finally, we substituted x = 2 into the derivative to find the specific rate of change at that point.

Remember, the quotient rule is your friend when dealing with functions in fractional form. And always take your time to simplify expressions – it makes everything much clearer! Whether you're studying calculus, working on a physics problem, or just curious about how things change, understanding derivatives and rates of change is super valuable. Keep practicing, and you'll become a pro in no time! Until next time, keep exploring the fascinating world of mathematics, guys!