Differentiate The Following Function:$\[ G(x) = \frac{5x - 2}{6x + 1} + X^3 \\]

Introduction

In calculus, differentiation is a fundamental concept that deals with the study of rates of change and slopes of curves. It is a crucial tool in various fields, including physics, engineering, and economics. In this article, we will focus on differentiating a given function, which is a combination of a rational function and a polynomial function. We will use the rules of differentiation to simplify the function and find its derivative.

The Given Function

The given function is:

${ g(x) = \frac{5x - 2}{6x + 1} + x^3 }$

This function consists of two parts: a rational function and a polynomial function. The rational function is:

${ \frac{5x - 2}{6x + 1} }$

And the polynomial function is:

${ x^3 }$

Differentiation Rules

To differentiate the given function, we will use the following rules:

• Power Rule: If ${ f(x) = x^n }$, then ${ f'(x) = nx^{n-1} }$
• Product Rule: If ${ f(x) = u(x)v(x) }$, then ${ f'(x) = u'(x)v(x) + u(x)v'(x) }$
• Quotient Rule: If ${ f(x) = \frac{u(x)}{v(x)} }$, then ${ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2} }$

Differentiating the Rational Function

To differentiate the rational function, we will use the quotient rule:

${ \frac{d}{dx} \left( \frac{5x - 2}{6x + 1} \right) = \frac{(5)(6x + 1) - (5x - 2)(6)}{(6x + 1)^2} }$

Simplifying the expression, we get:

${ \frac{30x + 5 - 30x + 12}{(6x + 1)^2} = \frac{17}{(6x + 1)^2} }$

Differentiating the Polynomial Function

To differentiate the polynomial function, we will use the power rule:

${ \frac{d}{dx} (x^3) = 3x^2 }$

Combining the Derivatives

Now that we have differentiated both parts of the function, we can combine the derivatives to find the derivative of the given function:

${ g'(x) = \frac{17}{(6x + 1)^2} + 3x^2 }$

Conclusion

In this article, we differentiated a given function that consisted of a rational function and a polynomial function. We used the quotient rule and the power rule to simplify the function and find its derivative. The resulting derivative is a combination of two functions, which can be used to analyze the behavior of the original function.

Applications of Differentiation

Differentiation has numerous applications in various fields, including:

• Physics: Differentiation is used to describe the motion of objects, including velocity and acceleration.
• Engineering: Differentiation is used to design and optimize systems, including electrical circuits and mechanical systems.
• Economics: Differentiation is used to analyze the behavior of economic systems, including supply and demand.

Final Thoughts

In conclusion, differentiation is a powerful tool that can be used to analyze and understand complex functions. By applying the rules of differentiation, we can simplify functions and find their derivatives, which can be used to analyze the behavior of the original function. Whether you are a student or a professional, understanding differentiation is essential for success in various fields.

References

• Calculus: Michael Spivak, "Calculus" (4th ed.), Publish or Perish, 2008.
• Differential Equations: Lawrence C. Evans, "Partial Differential Equations" (2nd ed.), American Mathematical Society, 2010.

Glossary

• Derivative: A measure of the rate of change of a function with respect to its input.
• Differentiation: The process of finding the derivative of a function.
• Quotient Rule: A rule for differentiating rational functions.
• Power Rule: A rule for differentiating polynomial functions.
  Differentiate the Given Function: A Comprehensive Analysis ===========================================================

Q&A: Differentiating the Given Function

In this article, we will continue to explore the concept of differentiation and provide answers to frequently asked questions related to differentiating the given function.

Q: What is the derivative of the rational function?

A: The derivative of the rational function is:

${ \frac{d}{dx} \left( \frac{5x - 2}{6x + 1} \right) = \frac{17}{(6x + 1)^2} }$

Q: How do I differentiate the polynomial function?

A: To differentiate the polynomial function, you can use the power rule:

${ \frac{d}{dx} (x^3) = 3x^2 }$

Q: What is the derivative of the given function?

A: The derivative of the given function is:

${ g'(x) = \frac{17}{(6x + 1)^2} + 3x^2 }$

Q: Can I use the product rule to differentiate the given function?

A: No, you cannot use the product rule to differentiate the given function. The product rule is used to differentiate functions of the form ${ f(x) = u(x)v(x) }$, but the given function is a combination of a rational function and a polynomial function.

Q: Can I use the quotient rule to differentiate the polynomial function?

A: No, you cannot use the quotient rule to differentiate the polynomial function. The quotient rule is used to differentiate rational functions, but the polynomial function is not a rational function.

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

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

• Forgetting to apply the power rule: Make sure to apply the power rule when differentiating polynomial functions.
• Forgetting to apply the quotient rule: Make sure to apply the quotient rule when differentiating rational functions.
• Not simplifying the expression: Make sure to simplify the expression after differentiating the function.

Q: How do I apply the rules of differentiation?

A: To apply the rules of differentiation, follow these steps:

1. Identify the type of function: Determine whether the function is a polynomial function, a rational function, or a combination of both.
2. Apply the appropriate rule: Apply the power rule, quotient rule, or product rule depending on the type of function.
3. Simplify the expression: Simplify the expression after differentiating the function.

Q: What are some real-world applications of differentiation?

A: Some real-world applications of differentiation include:

• Physics: Differentiation is used to describe the motion of objects, including velocity and acceleration.
• Engineering: Differentiation is used to design and optimize systems, including electrical circuits and mechanical systems.
• Economics: Differentiation is used to analyze the behavior of economic systems, including supply and demand.

Conclusion

In this article, we provided answers to frequently asked questions related to differentiating the given function. We also discussed some common mistakes to avoid when differentiating functions and provided some real-world applications of differentiation. Whether you are a student or a professional, understanding differentiation is essential for success in various fields.

References

• Calculus: Michael Spivak, "Calculus" (4th ed.), Publish or Perish, 2008.
• Differential Equations: Lawrence C. Evans, "Partial Differential Equations" (2nd ed.), American Mathematical Society, 2010.

Glossary

• Derivative: A measure of the rate of change of a function with respect to its input.
• Differentiation: The process of finding the derivative of a function.
• Quotient Rule: A rule for differentiating rational functions.
• Power Rule: A rule for differentiating polynomial functions.