Derivative Of F(x) = (-x^4 + 2x^2 + 1) / (x^3 - 9)

by Andrew McMorgan 51 views

Hey math enthusiasts! Today, we're diving into a calculus problem that might seem a bit intimidating at first glance, but trust me, we'll break it down step by step. We're going to find the derivative of the function f(x) = (-x^4 + 2x^2 + 1) / (x^3 - 9). This involves using the quotient rule, a fundamental concept in differential calculus. So, grab your calculators, sharpen your pencils, and let's get started!

Understanding the Quotient Rule

Before we jump into the problem, let's quickly recap the quotient rule. This rule is essential for finding the derivative of a function that is expressed as a fraction, where both the numerator and the denominator are functions of x. The quotient rule states that if we have a function f(x) = u(x) / v(x), then its derivative, denoted as f'(x), is given by:

f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2

Where:

  • u(x) is the numerator of the function.
  • v(x) is the denominator of the function.
  • u'(x) is the derivative of the numerator.
  • v'(x) is the derivative of the denominator.

In simpler terms, the derivative of a quotient is the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Got it? Great! Now, let's apply this to our specific problem.

Applying the Quotient Rule to Our Function

Okay, guys, let's get our hands dirty with the actual problem. Our function is f(x) = (-x^4 + 2x^2 + 1) / (x^3 - 9). To apply the quotient rule, we first need to identify our u(x) and v(x). Clearly:

  • u(x) = -x^4 + 2x^2 + 1
  • v(x) = x^3 - 9

Next, we need to find the derivatives of u(x) and v(x). This is where the power rule comes in handy. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). Let's apply this to find u'(x) and v'(x).

Finding u'(x)

u(x) = -x^4 + 2x^2 + 1

To find u'(x), we differentiate each term with respect to x:

  • The derivative of -x^4 is -4x^3 (using the power rule).
  • The derivative of 2x^2 is 4x (using the power rule).
  • The derivative of 1 is 0 (the derivative of a constant is always zero).

So, u'(x) = -4x^3 + 4x. Awesome!

Finding v'(x)

v(x) = x^3 - 9

Similarly, to find v'(x), we differentiate each term with respect to x:

  • The derivative of x^3 is 3x^2 (using the power rule).
  • The derivative of -9 is 0 (the derivative of a constant is zero).

Therefore, v'(x) = 3x^2. Fantastic! We've got our u(x), v(x), u'(x), and v'(x). Now we can plug these into the quotient rule formula.

Plugging into the Quotient Rule Formula

Remember the quotient rule? f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2. Let's substitute the values we found:

f'(x) = [(-4x^3 + 4x)(x^3 - 9) - (-x^4 + 2x^2 + 1)(3x^2)] / (x^3 - 9)^2

Okay, this looks a bit messy, but don't worry, we'll simplify it in the next step.

Simplifying the Expression

Now comes the fun part: simplifying the expression! This involves expanding the products in the numerator and then combining like terms. Let's tackle it step by step.

Expanding the Numerator

First, let's expand (-4x^3 + 4x)(x^3 - 9):

(-4x^3 + 4x)(x^3 - 9) = -4x^6 + 36x^3 + 4x^4 - 36x

Next, let's expand (-x^4 + 2x^2 + 1)(3x^2):

(-x^4 + 2x^2 + 1)(3x^2) = -3x^6 + 6x^4 + 3x^2

Now, let's put it all together in the numerator, remembering the minus sign between the two terms:

(-4x^6 + 36x^3 + 4x^4 - 36x) - (-3x^6 + 6x^4 + 3x^2)

Combining Like Terms

Now, let's combine the like terms in the numerator. Be careful with the signs!

-4x^6 + 36x^3 + 4x^4 - 36x + 3x^6 - 6x^4 - 3x^2

Combining the terms, we get:

-x^6 - 2x^4 + 36x^3 - 3x^2 - 36x

So, the simplified numerator is -x^6 - 2x^4 + 36x^3 - 3x^2 - 36x. Great job!

Expanding the Denominator

Now, let's expand the denominator, which is (x^3 - 9)^2. This means we need to multiply (x^3 - 9) by itself:

(x^3 - 9)^2 = (x^3 - 9)(x^3 - 9)

Expanding this, we get:

x^6 - 18x^3 + 81

So, the denominator is x^6 - 18x^3 + 81.

The Final Answer

Alright, we've done the hard work! Now we can write out the final answer. The derivative of f(x) = (-x^4 + 2x^2 + 1) / (x^3 - 9) is:

f'(x) = (-x^6 - 2x^4 + 36x^3 - 3x^2 - 36x) / (x^6 - 18x^3 + 81)

And there you have it! We've successfully found the derivative of the given function using the quotient rule. It might have seemed daunting at first, but by breaking it down into smaller steps and carefully applying the rules of calculus, we were able to solve it. Give yourself a pat on the back, guys! You've earned it.

Practice Makes Perfect

Remember, the key to mastering calculus is practice. The more problems you solve, the more comfortable you'll become with the concepts and techniques. So, don't be afraid to tackle more challenging problems. Try applying the quotient rule to other functions, and see if you can simplify the expressions. You might even want to use online derivative calculators to check your answers and make sure you're on the right track.

Further Exploration

If you're interested in learning more about derivatives and calculus, there are tons of resources available online and in libraries. You can explore topics like the chain rule, implicit differentiation, and applications of derivatives in various fields like physics, engineering, and economics. Calculus is a powerful tool, and the more you understand it, the more you'll be able to apply it to solve real-world problems.

Conclusion

So, that's how you find the derivative of f(x) = (-x^4 + 2x^2 + 1) / (x^3 - 9) using the quotient rule. I hope this explanation was helpful and easy to follow. Keep practicing, keep exploring, and most importantly, keep enjoying the fascinating world of mathematics! Until next time, happy calculating, folks!