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Introduction

In this article, we will explore the process of differentiating a complex function, specifically the function $y = x^2 + 8x + 6\sqrt{x+8}$. This function involves a square root term, which can make it challenging to differentiate. We will break down the problem into manageable steps and use various differentiation techniques to arrive at the final derivative.

Step 1: Identify the Function

The given function is $y = x^2 + 8x + 6\sqrt{x+8}$. This function consists of three terms: a quadratic term $x^2$, a linear term $8x$, and a square root term $6\sqrt{x+8}$.

Step 2: Differentiate the Quadratic Term

The derivative of the quadratic term $x^2$ is $2x$. This is a basic differentiation rule, where the derivative of $x^n$ is $nx^{n-1}$.

Step 3: Differentiate the Linear Term

The derivative of the linear term $8x$ is $8$. This is also a basic differentiation rule, where the derivative of $ax$ is $a$.

Step 4: Differentiate the Square Root Term

The derivative of the square root term $6\sqrt{x+8}$ requires the chain rule. We can rewrite the square root term as $6(x+8)^{\frac{1}{2}}$. The derivative of this term is $6 \cdot \frac{1}{2} (x+8)^{-\frac{1}{2}} \cdot 1$. Simplifying this expression, we get $\frac{3}{\sqrt{x+8}}$.

Step 5: Combine the Derivatives

Now that we have differentiated each term, we can combine the derivatives to get the final derivative of the function. The derivative of the quadratic term is $2x$, the derivative of the linear term is $8$, and the derivative of the square root term is $\frac{3}{\sqrt{x+8}}$. Therefore, the final derivative of the function is:

$$\frac{dy}{dx} = 2x + 8 + \frac{3}{\sqrt{x+8}}$$

Discussion

Differentiating a complex function like $y = x^2 + 8x + 6\sqrt{x+8}$ requires a step-by-step approach. We must identify the function, differentiate each term separately, and then combine the derivatives to get the final derivative. In this case, we used the chain rule to differentiate the square root term.

Conclusion

In conclusion, differentiating a complex function like $y = x^2 + 8x + 6\sqrt{x+8}$ requires careful attention to detail and a thorough understanding of differentiation rules. By breaking down the problem into manageable steps and using various differentiation techniques, we can arrive at the final derivative of the function.

Additional Resources

For more information on differentiation rules and techniques, please refer to the following resources:

• Calculus Textbook
• Mathway
• Khan Academy

Related Topics

• Differentiation rules
• Chain rule
• Square root function
• Quadratic function
• Linear function

FAQs

• Q: What is the derivative of the quadratic term $x^2$? A: The derivative of the quadratic term $x^2$ is $2x$.
• Q: What is the derivative of the linear term $8x$? A: The derivative of the linear term $8x$ is $8$.
• Q: What is the derivative of the square root term $6\sqrt{x+8}$? A: The derivative of the square root term $6\sqrt{x+8}$ is $\frac{3}{\sqrt{x+8}}$.
  Differentiating a Complex Function: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the process of differentiating a complex function, specifically the function $y = x^2 + 8x + 6\sqrt{x+8}$. We broke down the problem into manageable steps and used various differentiation techniques to arrive at the final derivative. In this article, we will provide a Q&A guide to help you better understand the process of differentiating complex functions.

Q: What is the derivative of the quadratic term $x^2$?

A: The derivative of the quadratic term $x^2$ is $2x$. This is a basic differentiation rule, where the derivative of $x^n$ is $nx^{n-1}$.

Q: What is the derivative of the linear term $8x$?

A: The derivative of the linear term $8x$ is $8$. This is also a basic differentiation rule, where the derivative of $ax$ is $a$.

Q: What is the derivative of the square root term $6\sqrt{x+8}$?

A: The derivative of the square root term $6\sqrt{x+8}$ requires the chain rule. We can rewrite the square root term as $6(x+8)^{\frac{1}{2}}$. The derivative of this term is $6 \cdot \frac{1}{2} (x+8)^{-\frac{1}{2}} \cdot 1$. Simplifying this expression, we get $\frac{3}{\sqrt{x+8}}$.

Q: How do I combine the derivatives of the quadratic, linear, and square root terms?

A: To combine the derivatives, we simply add them together. The derivative of the quadratic term is $2x$, the derivative of the linear term is $8$, and the derivative of the square root term is $\frac{3}{\sqrt{x+8}}$. Therefore, the final derivative of the function is:

$$\frac{dy}{dx} = 2x + 8 + \frac{3}{\sqrt{x+8}}$$

Q: What are some common mistakes to avoid when differentiating complex functions?

A: Some common mistakes to avoid when differentiating complex functions include:

• Forgetting to apply the chain rule when differentiating composite functions
• Not simplifying the derivative after applying the chain rule
• Not combining the derivatives of the individual terms correctly
• Not checking the domain of the function before differentiating

Q: How do I check the domain of the function before differentiating?

A: To check the domain of the function, we need to ensure that the function is defined for all values of $x$ in the given interval. In the case of the function $y = x^2 + 8x + 6\sqrt{x+8}$, the domain is all real numbers greater than or equal to $-8$. This is because the square root term is only defined for non-negative values of $x+8$.

Q: What are some real-world applications of differentiating complex functions?

A: Differentiating complex functions has many real-world applications, including:

• Physics: Differentiating complex functions is used to model the motion of objects in physics, such as the trajectory of a projectile or the vibration of a spring.
• Engineering: Differentiating complex functions is used to design and optimize systems, such as electronic circuits or mechanical systems.
• Economics: Differentiating complex functions is used to model economic systems and make predictions about future economic trends.

Conclusion

In conclusion, differentiating complex functions requires careful attention to detail and a thorough understanding of differentiation rules. By following the steps outlined in this article and avoiding common mistakes, you can successfully differentiate complex functions and apply them to real-world problems.

Additional Resources

For more information on differentiation rules and techniques, please refer to the following resources:

• Calculus Textbook
• Mathway
• Khan Academy

Related Topics

• Differentiation rules
• Chain rule
• Square root function
• Quadratic function
• Linear function

FAQs

• Q: What is the derivative of the quadratic term $x^2$? A: The derivative of the quadratic term $x^2$ is $2x$.
• Q: What is the derivative of the linear term $8x$? A: The derivative of the linear term $8x$ is $8$.
• Q: What is the derivative of the square root term $6\sqrt{x+8}$? A: The derivative of the square root term $6\sqrt{x+8}$ is $\frac{3}{\sqrt{x+8}}$.

Glossary

• Derivative: A measure of how a function changes as its input changes.
• Chain rule: A differentiation rule that allows us to differentiate composite functions.
• Composite function: A function that is composed of multiple functions.
• Domain: The set of all possible input values for a function.