Solving Complex Equations: A Step-by-Step Guide

Hey guys! Let's dive into some math problems. We're going to break down how to solve these equations step-by-step. Don't worry, it's not as scary as it looks! We'll tackle each problem, explaining the process clearly so you can understand it. We'll be using different techniques like the chain rule and other differentiation methods to find the solutions. So, grab your pencils and let's get started. Remember, practice makes perfect, so the more you work through these problems, the better you'll become! This guide will cover four different types of equations, each requiring a slightly different approach. We'll explore the use of the chain rule, product rule, and other differentiation techniques. Our goal is to make these equations understandable and help you build your problem-solving skills. So let's get started. First up, we'll deal with polynomial functions, followed by radical functions, and then a combination of both. Lastly, we will go over the division or the fraction of two equations, making sure we have everything covered! Keep in mind that understanding the principles behind each solution is key. It's not just about getting the right answer; it's about learning the process. Ready? Let's go!

Equation 1: y = (x² - x + 9)⁵

Alright, let's start with our first equation: y = (x² - x + 9)⁵. This is a classic example of a function where we need to apply the chain rule. The chain rule is super important when dealing with composite functions – functions within functions. In this case, we have a polynomial expression, (x² - x + 9), raised to the power of 5. Think of it like this: the outer function is something to the power of 5, and the inner function is the polynomial. We can think of this as a function of a function.

To solve this, we'll take the derivative. The chain rule states that the derivative of a composite function is the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function. Let's break that down. First, we take the derivative of the outer function, which is the power rule: bring down the exponent (5), keep the inside the same, and reduce the exponent by 1. So, we get 5 * (x² - x + 9)⁴. Next, we multiply this by the derivative of the inner function, which is x² - x + 9. The derivative of that is 2x - 1.

Putting it all together, the derivative of y = (x² - x + 9)⁵ is dy/dx = 5 * (x² - x + 9)⁴ * (2x - 1). And that's it! We have successfully applied the chain rule to find the derivative of this equation. Remember to always apply the chain rule when there is a function within a function. The chain rule is fundamental and understanding how to apply it is super important. We used the chain rule, a core concept in calculus, to efficiently find the derivative. Always remember the chain rule; it's your friend when dealing with composite functions! This is an important concept in calculus and is used to find derivatives of composite functions. Keep this in mind, and you will do great.

Equation 2: y = √(3x² + 9x - 2)³

Now, let's move on to our second equation: y = √(3x² + 9x - 2)³. This one looks a bit more complicated, but don't worry, we can handle it. This time, we have a square root of a polynomial expression raised to the power of 3. Here, we'll need to use both the chain rule and the power rule. First, let's rewrite the square root as a power of 1/2. So, y = (3x² + 9x - 2)^(3/2). Now it looks much easier, right? This is a composite function, so the chain rule will be our main tool. The outer function is something to the power of 3/2, and the inner function is the polynomial. We'll start by applying the power rule and then the chain rule.

Apply the power rule, bring down the exponent, and reduce it by 1, resulting in: (3/2) * (3x² + 9x - 2)^(1/2). Now we multiply that result by the derivative of the inner function. The derivative of 3x² + 9x - 2 is 6x + 9. Therefore, the complete derivative is dy/dx = (3/2) * (3x² + 9x - 2)^(1/2) * (6x + 9). Simplifying it a bit, we get dy/dx = (3/2) * √(3x² + 9x - 2) * (6x + 9).

Here, the chain rule was used along with the power rule to simplify the problem, so you just need to follow the formula. Now, remember that you can simplify the (6x + 9) by factoring out a 3 to get 3(2x + 3). The final answer can be written as dy/dx = (9/2) * √(3x² + 9x - 2) * (2x + 3). That looks much better. We've successfully found the derivative of this more complex equation by combining the power rule and the chain rule. The key is to break down the problem into smaller, manageable steps. Notice how we keep applying the chain rule because we're continuously dealing with functions within functions. Great job! Keep practicing, and it will become second nature.

Equation 3: y = 7 * ³√(x² + 6x - 1)

Okay, let's tackle our third equation: y = 7 * ³√(x² + 6x - 1). This equation involves a cube root and a constant multiplier. This time, we're going to work with a cube root. Just like with the square root, we can rewrite the cube root as a fractional exponent. So, we'll rewrite the equation as y = 7 * (x² + 6x - 1)^(1/3). The constant 7 will stay there. This way, it's easier to differentiate. We'll again use the chain rule, recognizing that we have a function within a function.

The outer function is something to the power of 1/3, and the inner function is our polynomial. Start by taking the derivative of the outer function. This means multiplying by the exponent (1/3), keeping the inside the same, and reducing the exponent by 1. Therefore, you get (1/3) * (x² + 6x - 1)^(-2/3). Now, multiply this result by the derivative of the inner function, which is 2x + 6. Don't forget the constant multiplier! The constant factor remains unaffected by the differentiation process.

So, the derivative of y = 7 * ³√(x² + 6x - 1) is dy/dx = 7 * (1/3) * (x² + 6x - 1)^(-2/3) * (2x + 6). Simplifying, we get dy/dx = (7/3) * (2x + 6) / (x² + 6x - 1)^(2/3). We put the negative exponent in the denominator. You can further simplify the answer, by taking out the 2 from the (2x + 6), which is 2(x+3). The final result becomes: dy/dx = (14 * (x + 3)) / (3 * (x² + 6x - 1)^(2/3)). Remember, pay close attention to the exponents and the signs. And always keep in mind that the chain rule is your best friend. Congrats! We have successfully taken the derivative of this equation.

Equation 4: y = (1 / (x² - 2x + 3)⁴)

Alright, let's finish with our last equation: y = 1 / (x² - 2x + 3)⁴. This time, we have a fraction. To make it easier to differentiate, let's rewrite this equation using a negative exponent: y = (x² - 2x + 3)^(-4). Now, we can see that this is another case where we need to apply the chain rule. The outer function is something to the power of -4, and the inner function is the polynomial x² - 2x + 3.

Apply the power rule to the outer function, bringing down the exponent (-4), keeping the inside the same, and reducing the exponent by 1. That gets us -4 * (x² - 2x + 3)^(-5). Then, multiply this by the derivative of the inner function, which is 2x - 2. So, the derivative of y = (1 / (x² - 2x + 3)⁴) is dy/dx = -4 * (x² - 2x + 3)^(-5) * (2x - 2).

We can simplify this by moving the term with the negative exponent to the denominator and also factoring out a 2 from the (2x - 2). This gives us dy/dx = (-8 * (x - 1)) / (x² - 2x + 3)⁵.

And there you have it! We've found the derivative of our fourth equation. By rewriting the equation and applying the chain rule, we made the problem much easier to solve. Great job completing all the exercises. Keep practicing, and you will master these techniques in no time. Congratulations!