Derivative Calculation: Y = (5x^2 + 6)(8x + 5 + 7/x)

by Alex Johnson 53 views

Let's dive into the process of finding the derivative of the function y = (5x^2 + 6)(8x + 5 + 7/x). This requires a solid understanding of calculus, specifically the product rule and power rule. We'll break down the steps to make it easy to follow.

Understanding the Problem

Before we start, it’s important to understand what a derivative is. In simple terms, the derivative of a function represents its instantaneous rate of change at a given point. In graphical terms, it gives us the slope of the tangent line to the curve at that point. For our function, y = (5x^2 + 6)(8x + 5 + 7/x), we want to find how y changes with respect to x.

Why is this important? Derivatives are fundamental in many areas of science and engineering. They help us optimize processes, understand rates of change, and model real-world phenomena. Whether you're calculating the velocity of a moving object or designing an efficient algorithm, derivatives play a crucial role.

Our function is a product of two expressions: (5x^2 + 6) and (8x + 5 + 7/x). This means we'll need to use the product rule, which states that the derivative of two functions multiplied together is given by:

(uv)' = u'v + uv'

Where:

  • u and v are functions of x
  • u' is the derivative of u with respect to x
  • v' is the derivative of v with respect to x

To apply the product rule, we'll first identify u and v, then find their individual derivatives. This will set us up for the next step, where we plug everything into the product rule formula. Make sure you're comfortable with the power rule as well, as we'll use it to differentiate the individual terms within u and v. The power rule states that the derivative of x^n is nx^(n-1).

Step-by-Step Solution

1. Identify u and v

Let's identify the two functions in our product:

  • u = 5x^2 + 6
  • v = 8x + 5 + 7/x = 8x + 5 + 7x^(-1) (Rewriting 7/x for easier differentiation)

2. Find the derivatives of u and v

Now we need to find u' and v'. We'll use the power rule and the constant rule (the derivative of a constant is 0) to do this.

Finding u' (the derivative of u):

u = 5x^2 + 6

u' = d/dx (5x^2 + 6)

Applying the power rule to 5x^2, we get 10x (because 2 * 5 * x^(2-1) = 10x). The derivative of the constant 6 is 0. Therefore:

u' = 10x + 0 = 10x

Finding v' (the derivative of v):

v = 8x + 5 + 7x^(-1)

v' = d/dx (8x + 5 + 7x^(-1))

Applying the power rule to 8x, we get 8 (because 1 * 8 * x^(1-1) = 8). The derivative of the constant 5 is 0. Applying the power rule to 7x^(-1), we get -7x^(-2) (because -1 * 7 * x^(-1-1) = -7x^(-2)). Therefore:

v' = 8 + 0 - 7x^(-2) = 8 - 7/x^2

3. Apply the product rule

Now that we have u, v, u', and v', we can apply the product rule:

(uv)' = u'v + uv'

Substitute the values we found:

(uv)' = (10x)(8x + 5 + 7/x) + (5x^2 + 6)(8 - 7/x^2)

4. Simplify the expression

Next, we need to simplify the expression. This involves expanding the terms and combining like terms.

(uv)' = 10x(8x + 5 + 7/x) + (5x^2 + 6)(8 - 7/x^2)

Expand the first term:

10x(8x + 5 + 7/x) = 80x^2 + 50x + 70

Expand the second term:

(5x^2 + 6)(8 - 7/x^2) = 40x^2 - 35 + 48 - 42/x^2 = 40x^2 + 13 - 42/x^2

Now combine the expanded terms:

(uv)' = 80x^2 + 50x + 70 + 40x^2 + 13 - 42/x^2

Combine like terms:

(uv)' = (80x^2 + 40x^2) + 50x + (70 + 13) - 42/x^2

(uv)' = 120x^2 + 50x + 83 - 42/x^2

So, the derivative of y = (5x^2 + 6)(8x + 5 + 7/x) is:

dy/dx = 120x^2 + 50x + 83 - 42/x^2

This is our final simplified derivative.

Common Mistakes to Avoid

When calculating derivatives, especially with the product rule, it's easy to make mistakes. Here are some common pitfalls to watch out for:

  • Forgetting the Product Rule: A common mistake is to simply differentiate each term separately and multiply the results. This is incorrect. Always remember to use the product rule when differentiating a product of two functions.
  • Incorrectly Applying the Power Rule: Make sure you correctly apply the power rule. Remember that the derivative of x^n is nx^(n-1). Pay close attention to negative exponents and fractions.
  • Sign Errors: Be very careful with signs, especially when dealing with negative terms or when expanding expressions. A small sign error can lead to a completely wrong answer.
  • Forgetting the Constant Rule: The derivative of a constant is always 0. Don't forget to apply this rule when differentiating terms that include constants.
  • Not Simplifying the Expression: While finding the derivative is the main goal, it's important to simplify your answer as much as possible. This makes it easier to work with the derivative in further calculations or applications.

By being aware of these common mistakes, you can increase your accuracy and confidence when calculating derivatives.

Alternative Methods and Tools

While we've walked through the manual calculation of the derivative, there are alternative methods and tools you can use to check your work or handle more complex functions.

1. Online Derivative Calculators

There are many online derivative calculators available that can quickly find the derivative of a function. These calculators are great for verifying your work or for handling functions that are too complex to differentiate by hand. Some popular options include Wolfram Alpha, Symbolab, and Desmos. Simply enter your function, and the calculator will provide the derivative, often along with step-by-step solutions.

2. Computer Algebra Systems (CAS)

CAS software like Mathematica, Maple, and SageMath are powerful tools for symbolic mathematics. They can handle a wide range of mathematical operations, including differentiation, integration, and equation solving. These tools are particularly useful for advanced calculus and for working with complex functions. CAS software can provide exact solutions and can also perform numerical calculations.

3. Numerical Differentiation

In cases where a function is difficult or impossible to differentiate analytically (i.e., using symbolic methods), numerical differentiation techniques can be used. These methods approximate the derivative at a point using finite differences. While numerical differentiation doesn't provide an exact formula for the derivative, it can give a good approximation, especially when dealing with real-world data or simulations.

4. Using Differentiation Rules and Tables

Another helpful approach is to become very familiar with basic differentiation rules and to use a table of derivatives. This can speed up the process and reduce the likelihood of errors. Common rules include the power rule, product rule, quotient rule, and chain rule, as well as the derivatives of common functions like trigonometric, exponential, and logarithmic functions.

By combining manual calculation with these alternative methods and tools, you can develop a strong understanding of differentiation and handle a wide range of problems effectively.

Conclusion

In this detailed walkthrough, we've successfully calculated the derivative of the function y = (5x^2 + 6)(8x + 5 + 7/x). We began by understanding the problem and identifying the need for the product rule. We then broke down the function into its components, found their individual derivatives, applied the product rule, and simplified the resulting expression. Along the way, we highlighted common mistakes to avoid and discussed alternative methods and tools for checking your work.

Derivatives are a crucial concept in calculus and have wide-ranging applications in various fields. By mastering the techniques and rules of differentiation, you'll be well-equipped to tackle complex problems and gain a deeper understanding of how functions change. Remember to practice regularly and use the resources available to you, such as online calculators and computer algebra systems, to reinforce your learning.

For further reading and a deeper dive into calculus, check out Khan Academy's Calculus Course, a great resource for learning and practicing calculus concepts.