Mastering Differentiation A Comprehensive Guide To Differentiating Y = 3 Log₃(x) - Cot(2x) + 3eˣ + 7/x² + Sin(2x)/cos(x)

Introduction

In the realm of calculus, differentiation stands as a fundamental operation, crucial for understanding rates of change and the behavior of functions. This article delves into the process of differentiating a complex function, breaking it down step by step to ensure clarity and comprehension. Our focus is on the function:

$ y = 3 \log _3 x-\cot 2 x+\frac{3}{e^{{-x}}+\frac{7}{x}2} +\frac{\sin 2 x}{\cos x} $

This function encompasses a variety of mathematical concepts, including logarithmic functions, trigonometric functions, exponential functions, and algebraic expressions. Each component requires a specific approach to differentiation, making this a comprehensive exercise in calculus. By meticulously examining each term and applying the appropriate differentiation rules, we will arrive at the derivative of the function with respect to x. This exploration will not only enhance your understanding of differentiation but also provide a practical guide for tackling similar complex functions. Understanding how to differentiate such functions is crucial in various fields, including physics, engineering, and economics, where rates of change and optimization are frequently analyzed. The ability to accurately differentiate complex functions allows for the modeling and prediction of real-world phenomena, making it an invaluable skill for students and professionals alike. Let's embark on this journey of differentiation, unraveling the intricacies of each term and mastering the techniques required to find the derivative of this multifaceted function. This article aims to serve as a detailed resource, providing both the steps and the explanations necessary to confidently approach differentiation problems.

Breaking Down the Function

To effectively differentiate the given function,

$ y = 3 \log _3 x - \cot 2x + \frac{3}{e^{-x}} + \frac{7}{x^2} + \frac{\sin 2x}{\cos x} $

we will address each term individually. This approach simplifies the process and allows us to apply the appropriate differentiation rules to each component. The function comprises five distinct terms: a logarithmic term ($3 \log _3 x$), a cotangent term ($\cot 2x$), an exponential term ($\frac{3}{e^{-x}}$), a power term ($\frac{7}{x^2}$), and a trigonometric term ($\frac{\sin 2x}{\cos x}$). Each of these terms has its own unique characteristics and requires a specific differentiation technique. For instance, the logarithmic term involves the differentiation of a logarithm with a base other than e, requiring a change of base formula or the application of the chain rule. The cotangent term necessitates the use of the derivative of the cotangent function and the chain rule due to the presence of $2x$. The exponential term, when simplified, becomes an exponential function with base e, which has a straightforward derivative. The power term can be differentiated using the power rule, while the trigonometric term requires the application of trigonometric identities and differentiation rules. By breaking down the function into these individual components, we can systematically differentiate each term and then combine the results to obtain the overall derivative. This method not only simplifies the differentiation process but also provides a clear and organized way to tackle complex functions. In the following sections, we will delve into the differentiation of each term, providing detailed explanations and step-by-step calculations to ensure a thorough understanding of the process. This methodical approach is essential for mastering differentiation and applying it to various mathematical problems.

1. Differentiating the Logarithmic Term: $3 \log _3 x$

The logarithmic term in our function is $3 \log _3 x$. To differentiate this, we need to recall the rule for differentiating logarithmic functions. The general rule for differentiating a logarithm with base b is:

$ \frac{d}{dx} [\log _b x] = \frac{1}{x \ln b} $

Applying this rule to our term, we get:

$ \frac{d}{dx} [3 \log _3 x] = 3 \cdot \frac{d}{dx} [\log _3 x] $

$ = 3 \cdot \frac{1}{x \ln 3} $

Thus, the derivative of $3 \log _3 x$ with respect to x is $\frac{3}{x \ln 3}$. This result highlights the importance of understanding the differentiation rules for logarithmic functions with different bases. The constant multiple rule allows us to bring the constant 3 outside the differentiation, simplifying the process. The derivative of $\log _3 x$ involves the natural logarithm of the base, which is a crucial component of the formula. This step-by-step approach ensures that we correctly apply the logarithmic differentiation rule and arrive at the accurate derivative. The ability to differentiate logarithmic functions is essential in various applications, including solving differential equations, analyzing growth rates, and modeling physical phenomena. A thorough understanding of the logarithmic differentiation rule is therefore a valuable asset in calculus and its applications. In the next sections, we will continue to differentiate the remaining terms of the function, building upon this foundation and applying other differentiation techniques.

2. Differentiating the Cotangent Term: $-\cot 2x$

Next, we address the cotangent term, which is $-\cot 2x$. To differentiate this, we need to remember the derivative of the cotangent function and apply the chain rule. The derivative of $\cot x$ is $-\csc^2 x$. Therefore, the derivative of $-\cot 2x$ can be found as follows:

$ \frac{d}{dx} [-\cot 2x] = - \frac{d}{dx} [\cot 2x] $

Applying the chain rule, we have:

$ = - [-\csc^2 (2x) \cdot \frac{d}{dx} (2x)] $

$ = \csc^2 (2x) \cdot 2 $

$ = 2 \csc^2 (2x) $

So, the derivative of $-\cot 2x$ with respect to x is $2 \csc^2 (2x)$. This differentiation demonstrates the application of the chain rule in conjunction with the derivative of a trigonometric function. The chain rule is crucial when differentiating composite functions, where one function is nested inside another. In this case, the function $2x$ is nested inside the cotangent function, necessitating the use of the chain rule. The derivative of the inner function, $2x$, is 2, which is then multiplied by the derivative of the outer function, $-\csc^2 (2x)$. The negative signs cancel out, resulting in a positive derivative. Understanding and correctly applying the chain rule is fundamental to differentiating a wide range of functions, particularly those involving trigonometric, exponential, and logarithmic components. This step-by-step breakdown illustrates how to systematically apply the chain rule and arrive at the accurate derivative of the cotangent term. In the subsequent sections, we will continue to differentiate the remaining terms of the function, further expanding our understanding of differentiation techniques.

3. Differentiating the Exponential Term: $\frac{3}{e^{-x}}$

Now, let's focus on the exponential term, $\frac{3}{e^{-x}}$. To differentiate this, we first simplify the term. Recall that $\frac{1}{e^{-x}} = e^x$. Therefore, our term becomes:

$ \frac{3}{e^{-x}} = 3e^x $

Now, we can easily differentiate this term. The derivative of $e^x$ is simply $e^x$. Applying this, we get:

$ \frac{d}{dx} [3e^x] = 3 \cdot \frac{d}{dx} [e^x] $

$ = 3e^x $

Thus, the derivative of $\frac{3}{e^{-x}}$ with respect to x is $3e^x$. This differentiation highlights the simplicity of differentiating exponential functions with base e. The exponential function $e^x$ has the unique property that its derivative is equal to itself, making it a fundamental function in calculus and its applications. The constant multiple rule allows us to bring the constant 3 outside the differentiation, further simplifying the process. By simplifying the term before differentiating, we avoided the need for more complex differentiation rules, such as the quotient rule. This step-by-step approach demonstrates the importance of simplifying expressions before differentiating, which can often lead to easier and more straightforward calculations. Exponential functions are prevalent in various fields, including physics, biology, and finance, making their differentiation a crucial skill. In the following sections, we will continue to differentiate the remaining terms of the function, building upon our understanding of differentiation techniques and applying them to different types of functions.

4. Differentiating the Power Term: $\frac{7}{x^2}$

Moving on, we consider the power term, $\frac{7}{x^2}$. To differentiate this term, we first rewrite it using negative exponents:

$ \frac{7}{x^2} = 7x^{-2} $

Now, we can apply the power rule for differentiation, which states that $\frac{d}{dx} [x^n] = nx^{n-1}$. Applying this rule, we get:

$ \frac{d}{dx} [7x^{-2}] = 7 \cdot \frac{d}{dx} [x^{-2}] $

$ = 7 \cdot (-2x^{-2-1}) $

$ = 7 \cdot (-2x^{-3}) $

$ = -14x^{-3} $

Rewriting this with a positive exponent, we have:

$ = -\frac{14}{x^3} $

Therefore, the derivative of $\frac{7}{x^2}$ with respect to x is $-\frac{14}{x^3}$. This differentiation demonstrates the application of the power rule, a fundamental rule in calculus for differentiating power functions. By rewriting the term with a negative exponent, we were able to directly apply the power rule, which simplifies the differentiation process. The power rule is widely applicable and is used to differentiate a variety of functions, making it an essential tool in calculus. The negative sign in the derivative indicates that the function is decreasing as x increases. This step-by-step breakdown illustrates how to systematically apply the power rule and arrive at the accurate derivative of the power term. In the subsequent sections, we will continue to differentiate the remaining terms of the function, further expanding our understanding of differentiation techniques and their applications.

5. Differentiating the Trigonometric Term: $\frac{\sin 2x}{\cos x}$

Finally, we differentiate the trigonometric term, $\frac{\sin 2x}{\cos x}$. To simplify this differentiation, we can use the trigonometric identity $\sin 2x = 2 \sin x \cos x$. Substituting this into our term, we get:

$ \frac{\sin 2x}{\cos x} = \frac{2 \sin x \cos x}{\cos x} $

We can cancel out the $\cos x$ terms, provided $\cos x \neq 0$:

$ = 2 \sin x $

Now, we differentiate $2 \sin x$. The derivative of $\sin x$ is $\cos x$. Therefore,

$ \frac{d}{dx} [2 \sin x] = 2 \cdot \frac{d}{dx} [\sin x] $

$ = 2 \cos x $

Thus, the derivative of $\frac{\sin 2x}{\cos x}$ with respect to x is $2 \cos x$. This differentiation highlights the importance of using trigonometric identities to simplify expressions before differentiating. By applying the double angle identity for sine, we were able to simplify the term significantly, making the differentiation process much easier. The derivative of the sine function is a fundamental concept in calculus and is used extensively in various applications. This step-by-step breakdown illustrates how to systematically simplify trigonometric expressions and differentiate them accurately. The ability to recognize and apply trigonometric identities is a valuable skill in calculus and is essential for solving a wide range of problems. In the next section, we will combine the derivatives of all the terms to find the overall derivative of the given function.

Combining the Derivatives

Having differentiated each term of the function,

$ y = 3 \log _3 x - \cot 2x + \frac{3}{e^{-x}} + \frac{7}{x^2} + \frac{\sin 2x}{\cos x} $

we now combine the individual derivatives to find the overall derivative of y with respect to x. We found the following derivatives for each term:

1. $ \frac{d}{dx} [3 \log _3 x] = \frac{3}{x \ln 3} $
2. $ \frac{d}{dx} [-\cot 2x] = 2 \csc^2 (2x) $
3. $ \frac{d}{dx} [\frac{3}{e^{-x}}] = 3e^x $
4. $ \frac{d}{dx} [\frac{7}{x^2}] = -\frac{14}{x^3} $
5. $ \frac{d}{dx} [\frac{\sin 2x}{\cos x}] = 2 \cos x $

Adding these derivatives together, we get the derivative of the entire function:

$ \frac{dy}{dx} = \frac{3}{x \ln 3} + 2 \csc^2 (2x) + 3e^x - \frac{14}{x^3} + 2 \cos x $

This final expression represents the derivative of the given function with respect to x. The process of combining the individual derivatives demonstrates the linearity of differentiation, which states that the derivative of a sum is the sum of the derivatives. This principle is fundamental in calculus and allows us to differentiate complex functions by breaking them down into simpler components. The overall derivative includes terms from various types of functions, including logarithmic, trigonometric, exponential, and power functions, showcasing the comprehensive nature of the differentiation process. This step-by-step approach ensures that we accurately combine the individual derivatives and arrive at the correct overall derivative. The ability to differentiate complex functions and combine the results is a crucial skill in calculus and its applications. In the conclusion, we will summarize the steps taken and highlight the key concepts and techniques used in this differentiation process.

Conclusion

In this comprehensive guide, we have successfully differentiated the complex function:

$ y = 3 \log _3 x - \cot 2x + \frac{3}{e^{-x}} + \frac{7}{x^2} + \frac{\sin 2x}{\cos x} $

with respect to x. We achieved this by systematically breaking down the function into individual terms and applying the appropriate differentiation rules to each. This methodical approach allowed us to handle the various types of functions present, including logarithmic, trigonometric, exponential, and power functions. We began by differentiating the logarithmic term, $3 \log _3 x$, using the rule for differentiating logarithms with a base other than e. We then differentiated the cotangent term, $-\cot 2x$, using the chain rule and the derivative of the cotangent function. For the exponential term, $\frac{3}{e^{-x}}$, we simplified it to $3e^x$ and then applied the derivative of $e^x$. The power term, $\frac{7}{x^2}$, was differentiated using the power rule after rewriting it with a negative exponent. Finally, we simplified the trigonometric term, $\frac{\sin 2x}{\cos x}$, using the double angle identity for sine and then differentiated the resulting expression. After obtaining the individual derivatives, we combined them to find the overall derivative of the function:

$ \frac{dy}{dx} = \frac{3}{x \ln 3} + 2 \csc^2 (2x) + 3e^x - \frac{14}{x^3} + 2 \cos x $

This process highlights the importance of understanding fundamental differentiation rules, such as the chain rule, power rule, and derivatives of trigonometric and exponential functions. It also emphasizes the value of simplifying expressions before differentiating, as this can often lead to easier calculations. The ability to differentiate complex functions is a crucial skill in calculus and has wide-ranging applications in various fields, including physics, engineering, and economics. By mastering the techniques presented in this guide, you can confidently approach similar differentiation problems and apply them to real-world scenarios. This article serves as a valuable resource for students and professionals alike, providing a clear and detailed explanation of the differentiation process for a complex function.