Derivative Of F(x) = (8x^2-6)/(2x^2+4): Evaluation At X=11,-3

Hey Plastik Magazine readers! Today, we're diving into the exciting world of calculus to tackle a problem that might seem a bit daunting at first, but trust me, it's totally manageable. We're going to explore how to find the derivative of a rational function and then evaluate it at specific points. So, grab your thinking caps, and let's get started!

Understanding the Function

First, let's get acquainted with our function: f(x) = (8x^2 - 6) / (2x^2 + 4). This is a rational function, which simply means it's a fraction where both the numerator and the denominator are polynomials. To find its derivative, we'll need to use a special rule called the quotient rule. The quotient rule is a formula that helps us find the derivative of any function that's expressed as a ratio of two other functions. It might sound complicated, but we'll break it down step by step.

Applying the Quotient Rule

Now, let's roll up our sleeves and apply the quotient rule. If we have a function h(x) that's defined as 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) = 8x^2 - 6 and v(x) = 2x^2 + 4. So, the first thing we need to do is find the derivatives of u(x) and v(x). Remember, the derivative of x^n is nx^(n-1), and the derivative of a constant is zero. Therefore, u'(x) = 16x and v'(x) = 4x. Now we have all the pieces we need to plug into the quotient rule formula. Let's do it! Plugging in, we get f'(x) = [(16x)(2x^2 + 4) - (8x^2 - 6)(4x)] / (2x^2 + 4)^2. Next, we need to simplify this expression. First, expand the terms in the numerator. We get (16x)(2x^2 + 4) = 32x^3 + 64x and (8x^2 - 6)(4x) = 32x^3 - 24x. So, the numerator becomes (32x^3 + 64x) - (32x^3 - 24x). Notice that the 32x^3 terms cancel out, which simplifies things nicely. What we're left with in the numerator is 64x - (-24x) = 64x + 24x = 88x. Now let's look at the denominator, which is (2x^2 + 4)^2. Expanding this gives us (2x^2 + 4)(2x^2 + 4) = 4x^4 + 16x^2 + 16. So, our derivative is now f'(x) = 88x / (4x^4 + 16x^2 + 16). We can simplify this further by noticing that each term in the denominator is divisible by 4. Dividing each term by 4, we get f'(x) = 88x / [4(x^4 + 4x^2 + 4)]. Then, we can simplify by canceling the common factor of 4, resulting in f'(x) = 22x / (x^4 + 4x^2 + 4). The denominator x^4 + 4x^2 + 4 looks like a perfect square trinomial. Specifically, it can be factored as (x^2 + 2)^2. Therefore, we can rewrite the derivative as f'(x) = 22x / (x^2 + 2)^2. This is the simplified form of the derivative of our original function f(x).

Evaluating at x = 11

Alright, we've found the derivative, which is f'(x) = 22x / (x^2 + 2)^2. Now, let's plug in x = 11 to find f'(11). This means we replace every 'x' in the derivative equation with '11'. So, f'(11) = 22 * 11 / (11^2 + 2)^2. First, let's calculate 11^2, which is 121. Then, we add 2 to that, so 121 + 2 = 123. Now we have f'(11) = 22 * 11 / (123)^2. Let's calculate the numerator first: 22 * 11 = 242. Next, we need to square 123, which means 123 * 123. This equals 15129. So, now we have f'(11) = 242 / 15129. This fraction doesn't simplify nicely, so we can leave it as is or convert it to a decimal. To convert it to a decimal, we divide 242 by 15129, which gives us approximately 0.016. Therefore, f'(11) is approximately 0.016. That’s the value of the derivative of our function at the point x = 11. It tells us the instantaneous rate of change of the function at that specific point. In graphical terms, it's the slope of the tangent line to the curve of the function at x = 11.

Evaluating at x = -3

Now, let's tackle the second part of our problem: evaluating the derivative at x = -3. We're going to use the same derivative we found earlier, which is f'(x) = 22x / (x^2 + 2)^2. This time, we'll replace every 'x' with '-3'. So, we have f'(-3) = 22 * (-3) / ((-3)^2 + 2)^2. First, let's deal with the numerator. 22 multiplied by -3 is -66, so the numerator is -66. Now, let's move on to the denominator. We have (-3)^2, which means -3 multiplied by -3. A negative times a negative is a positive, so (-3)^2 is 9. Then, we add 2 to that, giving us 9 + 2 = 11. Now we have f'(-3) = -66 / (11)^2. Next, we square 11, which is 11 * 11 = 121. So, our expression is now f'(-3) = -66 / 121. This fraction can be simplified because both -66 and 121 are divisible by 11. Dividing both the numerator and the denominator by 11, we get -66 / 11 = -6 and 121 / 11 = 11. Therefore, the simplified fraction is -6 / 11. So, f'(-3) = -6 / 11. This is the value of the derivative of our function at the point x = -3. Again, this value represents the slope of the tangent line to the curve of the function at this specific point. The negative sign indicates that the function is decreasing at x = -3. To get a decimal approximation, we can divide -6 by 11, which gives us approximately -0.545. Therefore, f'(-3) is approximately -0.545.

Conclusion

So, there you have it! We've successfully navigated the process of finding the derivative of a rational function using the quotient rule and evaluated it at two specific points. Remember, the derivative gives us valuable information about the rate of change of a function, and evaluating it at specific points helps us understand the function's behavior at those locations. I hope this breakdown has been helpful and has given you a better understanding of how to tackle these kinds of problems. Keep practicing, and you'll become a calculus pro in no time! Until next time, keep exploring the fascinating world of mathematics!