Differentiating Ln(x) / Sec(x) Using The Quotient Rule Step-by-Step

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The quotient rule is a fundamental concept in calculus that allows us to find the derivative of a function that is expressed as the quotient of two other functions. In simpler terms, if you have a function y that is defined as u(x) / v(x), where u(x) and v(x) are differentiable functions, then the derivative of y with respect to x, denoted as dy/dx, can be determined using the quotient rule formula. This rule is particularly useful when dealing with functions that involve fractions where both the numerator and the denominator are functions of the variable x. Mastering the quotient rule is crucial for various applications in calculus and related fields, including optimization problems, rate of change calculations, and curve sketching. Understanding its application enables us to analyze and model real-world phenomena more effectively, making it an indispensable tool in mathematical analysis.

The formula for the quotient rule is given by:

dydx=v(x)⋅u′(x)−u(x)⋅v′(x)[v(x)]2\frac{dy}{dx} = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}

Where:

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

This formula might look intimidating at first, but it becomes straightforward with practice. The key is to correctly identify u(x) and v(x) and then find their respective derivatives. The formula essentially states that the derivative of the quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator. This structured approach helps in systematically finding the derivative of complex quotient functions.

In this article, we will apply the quotient rule to differentiate the function:

y=ln⁡xsec⁡xy = \frac{\ln x}{\sec x}

This example will illustrate how to apply the quotient rule step-by-step, providing a clear understanding of the process involved in differentiating such functions. By breaking down the problem into manageable parts, we can effectively use the quotient rule to find the derivative and gain confidence in tackling similar calculus problems.

Step-by-Step Differentiation of y = ln(x) / sec(x)

To differentiate the function y = ln(x) / sec(x) using the quotient rule, we need to systematically apply the formula and break down the process into manageable steps. This will allow us to understand each component of the rule and how it contributes to the final result. The quotient rule, as we've discussed, is essential for differentiating functions that are expressed as a fraction where both the numerator and the denominator are functions of x. In this particular case, our numerator is ln(x), the natural logarithm of x, and our denominator is sec(x), the secant function of x. Both of these functions have well-defined derivatives, which we will need to apply the quotient rule effectively.

The step-by-step process involves first identifying the functions u(x) and v(x), then finding their derivatives u'(x) and v'(x), and finally substituting these into the quotient rule formula. This methodical approach not only helps in solving the problem accurately but also enhances understanding of the underlying principles of differentiation. Let's delve into the details of each step to ensure a clear and comprehensive understanding of the differentiation process.

1. Identify u(x) and v(x)

First, we identify the numerator and denominator functions:

  • u(x) = ln(x)
  • v(x) = sec(x)

2. Find u'(x) and v'(x)

Next, we find the derivatives of u(x) and v(x):

  • The derivative of u(x) = ln(x) is:

    u′(x)=1xu'(x) = \frac{1}{x}

The derivative of the natural logarithm is a fundamental result in calculus. It states that the rate of change of the natural logarithm function at any point x is inversely proportional to x. This derivative is widely used in various applications, including logarithmic differentiation, integration, and solving differential equations. Understanding this basic derivative is crucial for dealing with more complex functions involving logarithms.

  • The derivative of v(x) = sec(x) is:

    v′(x)=sec⁡xtan⁡xv'(x) = \sec x \tan x

Finding the derivative of sec(x) involves understanding trigonometric derivatives and the chain rule. The derivative of sec(x) is sec(x)tan(x), which is a standard result in calculus. This derivative is essential when dealing with trigonometric functions in quotient rule applications and other calculus problems. Mastering this derivative helps in solving problems involving trigonometric rates of change and optimization.

3. Apply the Quotient Rule

Now, we apply the quotient rule formula:

dydx=v(x)⋅u′(x)−u(x)⋅v′(x)[v(x)]2\frac{dy}{dx} = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}

Substitute u(x), v(x), u'(x), and v'(x) into the formula:

dydx=sec⁡x⋅1x−ln⁡x⋅sec⁡xtan⁡x(sec⁡x)2\frac{dy}{dx} = \frac{\sec x \cdot \frac{1}{x} - \ln x \cdot \sec x \tan x}{(\sec x)^2}

The application of the quotient rule formula is a critical step in differentiating quotient functions. The formula provides a structured way to combine the original functions and their derivatives to find the derivative of the entire quotient. Accurate substitution and simplification are essential to arrive at the correct result. This step demonstrates the practical use of the quotient rule in a specific example and reinforces the importance of following the formula precisely.

Simplify the Result

After applying the quotient rule, the next crucial step is to simplify the expression. Simplifying the result not only makes it easier to understand but also facilitates further calculations or analysis that might be required. The complexity of the expression obtained after applying the quotient rule often necessitates simplification to arrive at a more manageable form. This simplification typically involves algebraic manipulations, such as factoring, canceling common terms, and using trigonometric identities to reduce the expression to its simplest form. The simplified derivative provides clearer insights into the behavior of the original function and is more convenient for use in subsequent calculations or applications.

In this particular case, simplifying the derivative of y = ln(x) / sec(x) involves several steps, including factoring out common terms and applying trigonometric identities. These steps are essential to reduce the expression to a more concise and meaningful form. The simplified result will allow us to analyze the rate of change of the function more effectively and use it in further mathematical operations.

dydx=sec⁡x⋅1x−ln⁡x⋅sec⁡xtan⁡x(sec⁡x)2\frac{dy}{dx} = \frac{\sec x \cdot \frac{1}{x} - \ln x \cdot \sec x \tan x}{(\sec x)^2}

To simplify, we can factor out sec x from the numerator:

dydx=sec⁡x(1x−ln⁡xtan⁡x)(sec⁡x)2\frac{dy}{dx} = \frac{\sec x (\frac{1}{x} - \ln x \tan x)}{(\sec x)^2}

Now, cancel out one sec x from the numerator and denominator:

dydx=1x−ln⁡xtan⁡xsec⁡x\frac{dy}{dx} = \frac{\frac{1}{x} - \ln x \tan x}{\sec x}

Further Simplification (Optional)

We can further simplify the expression by multiplying both the numerator and the denominator by x to get rid of the fraction in the numerator:

dydx=1−xln⁡xtan⁡xxsec⁡x\frac{dy}{dx} = \frac{1 - x \ln x \tan x}{x \sec x}

This is the simplified form of the derivative. Simplifying the expression after applying the quotient rule is crucial for clarity and ease of use in further calculations. The simplified form of the derivative provides a more concise representation of the rate of change of the function. In this example, we factored out common terms and canceled them to reduce the complexity of the expression. Further simplification involved multiplying the numerator and the denominator by x to eliminate the fraction within the numerator, resulting in a cleaner and more manageable form. The simplified derivative is easier to interpret and use in subsequent mathematical operations, such as finding critical points or analyzing the function's behavior.

Final Answer

Thus, the derivative of y = ln(x) / sec(x) is:

dydx=1−xln⁡xtan⁡xxsec⁡x\frac{dy}{dx} = \frac{1 - x \ln x \tan x}{x \sec x}

The final answer represents the result of the differentiation process, where we have successfully applied the quotient rule and simplified the expression. This derivative gives us the rate of change of the original function, y = ln(x) / sec(x), with respect to x. The final answer is crucial for various applications, such as finding critical points, analyzing the function's behavior, and solving related mathematical problems. It is essential to present the final answer in a clear and concise form, making it easy to interpret and use in further calculations. In this case, the derivative provides valuable insights into how the function changes and behaves, making it a significant outcome of the differentiation process.

Conclusion

In conclusion, the quotient rule is a powerful tool in calculus for differentiating functions that are expressed as the quotient of two other functions. By following a systematic approach, we can effectively apply the quotient rule and simplify the resulting expressions. The step-by-step process, which includes identifying u(x) and v(x), finding their derivatives, applying the quotient rule formula, and simplifying the result, is essential for accurate differentiation. The quotient rule is not only a fundamental concept in calculus but also has practical applications in various fields, including physics, engineering, and economics, where rate of change and optimization problems are common. Mastering the quotient rule allows us to tackle complex differentiation problems with confidence and provides a solid foundation for further studies in calculus and related areas.

In this article, we demonstrated the application of the quotient rule to the function y = ln(x) / sec(x). We walked through each step, from identifying the functions to simplifying the final derivative. This example provides a clear understanding of how to use the quotient rule effectively and reinforces the importance of simplification in calculus problems. The ability to differentiate quotient functions is a valuable skill, enabling us to analyze and solve a wide range of mathematical problems.