Finding The Derivative Of Y = (x^6/12) Ln(x) - (x^6/72) A Calculus Guide
In the realm of calculus, finding derivatives is a fundamental operation. It allows us to understand the rate at which a function changes. In this article, we embark on a journey to find the derivative of the function $y = \frac{x^6}{12} \ln(x) - \frac{x^6}{72}$ with respect to $x$. This process will involve applying key differentiation rules and algebraic manipulation to arrive at the final result. Let's dive in!
Understanding the Foundation: Differentiation Rules
Before we plunge into the specifics of our problem, it's crucial to lay the groundwork by revisiting the essential differentiation rules that will serve as our guiding principles. These rules are the bedrock of differential calculus, providing us with the tools to tackle a wide array of functions. Among the most pertinent rules for our task are the power rule, the product rule, and the constant multiple rule. The power rule, a cornerstone of differentiation, dictates that the derivative of $x^n$ is $nx^{n-1}$, where $n$ is any real number. This rule empowers us to handle terms involving variables raised to powers. The product rule comes into play when we encounter functions that are products of other functions. It asserts that the derivative of $u(x)v(x)$ is $u'(x)v(x) + u(x)v'(x)$, where $u'(x)$ and $v'(x)$ represent the derivatives of $u(x)$ and $v(x)$, respectively. This rule is indispensable for differentiating expressions involving the multiplication of two functions. Lastly, the constant multiple rule simplifies the differentiation of terms multiplied by constants. It states that the derivative of $cf(x)$ is $cf'(x)$, where $c$ is a constant and $f'(x)$ is the derivative of $f(x)$. This rule allows us to extract constants from differentiation operations, making the process more manageable. Armed with these fundamental rules, we are well-equipped to embark on our quest to find the derivative of the given function. These rules are not mere formulas; they are the very essence of how we quantify change and understand the behavior of functions.
Applying the Product Rule and Power Rule
In this section, we'll dissect the given function, $y = \frac{x^6}{12} \ln(x) - \frac{x^6}{72}$, and strategically apply the product rule and power rule to unravel its derivative. Our function comprises two primary terms, the first being a product of $ \frac{x^6}{12} $ and $ \ln(x) $, and the second being a simple power function, $ \frac{x^6}{72} $. The presence of the product $ \frac{x^6}{12} \ln(x) $ immediately signals the need for the product rule. To effectively employ this rule, we'll designate $ u(x) = \frac{x^6}{12} $ and $ v(x) = \ln(x) $. The next step involves finding the derivatives of these individual components. Applying the power rule to $ u(x) = \frac{x^6}{12} $, we obtain $ u'(x) = \frac{6x^5}{12} = \frac{x^5}{2} $. The derivative of $ \ln(x) $, a fundamental result in calculus, is simply $ \frac{1}{x} $, thus $ v'(x) = \frac{1}{x} $. Now, with $ u(x) $, $ v(x) $, $ u'(x) $, and $ v'(x) $ in hand, we can invoke the product rule, which states that the derivative of $ u(x)v(x) $ is $ u'(x)v(x) + u(x)v'(x) $. Substituting our calculated derivatives, we get the derivative of $ \frac{x^6}{12} \ln(x) $ as $ \frac{x^5}{2} \ln(x) + \frac{x^6}{12} \cdot \frac{1}{x} $. Simplifying this expression yields $ \frac{x^5}{2} \ln(x) + \frac{x^5}{12} $. For the second term in our original function, $ -\frac{x^6}{72} $, we again wield the power rule. The derivative of $ -\frac{x^6}{72} $ is $ -\frac{6x^5}{72} = -\frac{x^5}{12} $. With these derivatives in hand, we are now poised to combine them and arrive at the complete derivative of our function.
Combining the Results and Simplifying
Having navigated the intricacies of the product rule and power rule, we now stand at the crucial juncture of combining our intermediate results and simplifying the expression to reveal the derivative of $y$ with respect to $x$. We've meticulously calculated the derivative of the first term, $ \frac{x^6}{12} \ln(x) $, as $ \frac{x^5}{2} \ln(x) + \frac{x^5}{12} $, and the derivative of the second term, $ -\frac{x^6}{72} $, as $ -\frac{x^5}{12} $. To obtain the complete derivative, we simply add these two results together. This yields the expression $ \frac{dy}{dx} = \frac{x^5}{2} \ln(x) + \frac{x^5}{12} - \frac{x^5}{12} $. Upon close examination, we observe a delightful simplification opportunity. The terms $ \frac{x^5}{12} $ and $ -\frac{x^5}{12} $ gracefully cancel each other out, leaving us with a more concise and elegant expression. This cancellation not only simplifies our result but also underscores the beauty of mathematical operations, where terms can sometimes coalesce and vanish, revealing a more fundamental structure. After this simplification, our derivative takes the form $ \frac{dy}{dx} = \frac{x^5}{2} \ln(x) $. This is the derivative of $y$ with respect to $x$, a testament to our methodical application of differentiation rules and algebraic manipulation. This final expression encapsulates the rate at which $y$ changes as $x$ varies, providing valuable insights into the function's behavior. It's a tangible outcome of our journey through calculus, a reward for our diligent exploration of differentiation techniques.
Final Answer: The Derivative of y
After a journey through the realms of differentiation rules, product rule applications, and meticulous simplification, we arrive at the final answer: the derivative of $y$ with respect to $x$. Our initial function, $y = \fracx^6}{12} \ln(x) - \frac{x^6}{72}$, has been transformed through the process of differentiation, revealing its rate of change. We carefully applied the product rule to the term $ \frac{x^6}{12} \ln(x) $, recognizing it as a product of two functions. The power rule played a crucial role in differentiating both $ x^6 $ terms. Through these operations, we meticulously calculated the individual derivatives and then combined them, paving the way for simplification. The terms $ \frac{x^5}{12} $ and $ -\frac{x^5}{12} $ gracefully canceled each other out, a testament to the elegance of mathematical simplification. This cancellation streamlined our expression, leading us to a more concise and insightful result. The culmination of our efforts is the derivative{dx} = \frac{x^5}{2} \ln(x)$. This expression encapsulates the rate at which $y$ changes in response to variations in $x$. It's a tangible manifestation of the function's dynamic behavior, providing us with a powerful tool for analysis and prediction. This final answer is not merely a result; it's a testament to the power of calculus and the methodical application of its principles. It represents a deeper understanding of the function's characteristics and its place within the broader mathematical landscape.
In conclusion, by meticulously applying the product rule, power rule, and simplifying the resulting expression, we have successfully determined that the derivative of $y = \frac{x^6}{12} \ln(x) - \frac{x^6}{72}$ with respect to $x$ is $\frac{x^5}{2} \ln(x)$. This exercise demonstrates the power of calculus in understanding the behavior of functions.
