Derivative Of Y = 22 + Ln(x³) Step-by-Step Solution

Introduction to Derivatives

In the realm of calculus, derivatives stand as fundamental tools for understanding rates of change and the behavior of functions. At its core, a derivative measures the instantaneous rate at which a function's output changes with respect to its input. This concept finds extensive applications across various fields, from physics and engineering to economics and computer science. Understanding derivatives allows us to analyze how quantities evolve, optimize systems, and model real-world phenomena with greater precision. The derivative of a function at a specific point represents the slope of the tangent line to the function's graph at that point, offering a visual and intuitive interpretation of its rate of change. In essence, derivatives provide a powerful lens through which we can examine the dynamic nature of functions and their applications.

The process of finding a derivative, known as differentiation, involves applying specific rules and techniques depending on the function's form. For polynomial functions, the power rule simplifies differentiation, while trigonometric functions necessitate knowledge of their respective derivative patterns. The chain rule becomes indispensable when dealing with composite functions, where one function is nested within another. Furthermore, implicit differentiation enables us to find derivatives of functions defined implicitly, where the dependent variable is not explicitly isolated. Mastery of these differentiation techniques equips us to tackle a wide array of derivative problems, paving the way for deeper insights into the behavior of functions and their applications in diverse contexts. For logarithmic functions, the rules for differentiation involve the natural logarithm and its properties. These rules, combined with other differentiation techniques, allow us to effectively analyze and manipulate expressions involving logarithms.

Problem Statement: y = 22 + ln(x³)

Our primary goal is to determine the derivative of the given function, y = 22 + ln(x³). This expression combines a constant term (22) with a natural logarithm function of a power of x (ln(x³)). To find the derivative, we will utilize a combination of differentiation rules, including the constant rule, the chain rule, and the power rule for logarithms. By systematically applying these rules, we can break down the expression into manageable components and arrive at the derivative, which will represent the instantaneous rate of change of y with respect to x. Understanding the derivative of this function allows us to analyze its behavior, such as its increasing or decreasing intervals, concavity, and critical points. This information is valuable in various applications, including optimization problems, curve sketching, and modeling real-world phenomena where logarithmic relationships are involved. The constant term, 22, will have a derivative of zero, while the logarithmic term will require careful application of the chain rule and the power rule of logarithms to find its derivative. By combining these results, we can determine the complete derivative of the given function.

Step-by-Step Solution

To find the derivative y' of the function y = 22 + ln(x³), we will proceed step-by-step, applying relevant differentiation rules.

Step 1: Apply the Sum Rule

The sum rule states that the derivative of a sum of functions is the sum of their derivatives. In this case, we have y = 22 + ln(x³). Thus, we can write:

y' = d/dx (22) + d/dx (ln(x³))

Step 2: Differentiate the Constant Term

The derivative of a constant is always zero. Therefore,

d/dx (22) = 0

Step 3: Apply the Power Rule of Logarithms

Before differentiating the logarithmic term, we can simplify it using the power rule of logarithms, which states that ln(a^b) = b * ln(a). Applying this rule to our expression, we get:

ln(x³) = 3 * ln(x)

So, our function now becomes:

y = 22 + 3 * ln(x)

And its derivative:

y' = 0 + d/dx (3 * ln(x))

Step 4: Differentiate the Logarithmic Term

The derivative of ln(x) with respect to x is 1/x. Also, we can use the constant multiple rule, which states that d/dx (c * f(x)) = c * d/dx (f(x)), where c is a constant. Applying these rules, we have:

d/dx (3 * ln(x)) = 3 * d/dx (ln(x)) = 3 * (1/x) = 3/x

Step 5: Combine the Results

Now, we combine the derivatives of the constant term and the logarithmic term:

y' = 0 + 3/x

Therefore, the derivative of y = 22 + ln(x³) is:

y' = 3/x

Alternative Approach: Chain Rule

Another way to find the derivative of y = 22 + ln(x³) is by directly applying the chain rule to the ln(x³) term. The chain rule states that if we have a composite function y = f(g(x)), then its derivative is y' = f'(g(x)) * g'(x).

In our case, we can consider f(u) = ln(u) and g(x) = x³. Then, f'(u) = 1/u and g'(x) = 3x². Applying the chain rule,

d/dx (ln(x³)) = f'(g(x)) * g'(x) = (1/x³) * 3x² = 3x²/x³ = 3/x

This approach directly differentiates ln(x³) without simplifying it using the power rule of logarithms first. The rest of the steps are the same as in the previous solution.

Conclusion

In conclusion, by applying the rules of differentiation, including the sum rule, the constant rule, the power rule of logarithms, and the chain rule, we have successfully found the derivative of the function y = 22 + ln(x³). The derivative, y' = 3/x, represents the rate of change of the function y with respect to x. This derivative is valuable for further analysis of the function's behavior, such as finding critical points, intervals of increase and decrease, and concavity. The derivative provides essential insights into the function's behavior and can be used for various applications, including optimization problems and curve sketching. The systematic application of differentiation rules, as demonstrated in this step-by-step solution, is crucial for mastering calculus and its applications in various fields.

Understanding the derivative y' = 3/x also allows us to analyze the slope of the tangent line to the curve y = 22 + ln(x³) at any point x. For positive values of x, the derivative is positive, indicating that the function is increasing. As x approaches infinity, the derivative approaches zero, suggesting that the rate of change becomes smaller. This information can be used to sketch the graph of the function and identify key features such as asymptotes and local extrema. Moreover, the derivative can be used to solve optimization problems, such as finding the maximum or minimum value of the function within a given interval. The derivative is a powerful tool in calculus, providing insights into the behavior of functions and their applications in various contexts.