Rewrite And Differentiate Log9(x^4 - 3x - 1)
Introduction
The challenge at hand involves a function, f(x) = log9(x^4 - 3x - 1), and the task is to rewrite it using logarithmic properties before differentiating. This approach often simplifies the differentiation process, making it more manageable and less prone to errors. The problem hints at utilizing the change of base formula for logarithms and then applying differentiation rules. Let's delve into the step-by-step process of rewriting the function and subsequently finding its derivative. We'll explore the underlying mathematical principles and demonstrate how these techniques can be applied to similar problems. Understanding these concepts is crucial for mastering calculus and its applications in various fields.
Rewriting the Function
The initial function is given as f(x) = log9(x^4 - 3x - 1). To rewrite this, we'll employ the change of base formula for logarithms. This formula states that logb(a) = ln(a) / ln(b), where ln represents the natural logarithm (logarithm to the base e). Applying this formula to our function, we get:
f(x) = ln(x^4 - 3x - 1) / ln(9)
Here, we've transformed the original logarithm with base 9 into a natural logarithm divided by the natural logarithm of 9. This form is often more convenient for differentiation because the natural logarithm has well-established differentiation rules. The denominator, ln(9), is a constant, which will simplify the differentiation process further. This step is a crucial preparatory stage, making the subsequent differentiation smoother and more accurate. By changing the base to the natural logarithm, we align the function with standard calculus techniques and formulas, paving the way for a straightforward derivative calculation. The rewritten function now clearly shows the structure needed for applying the quotient rule or constant multiple rule in differentiation.
Differentiating the Rewritten Function
Now that we have rewritten the function as f(x) = ln(x^4 - 3x - 1) / ln(9), we can proceed with differentiation. Recall that ln(9) is a constant, so we can treat it as a coefficient. The derivative of f(x), denoted as f'(x), can be found using the chain rule. The chain rule states that if we have a composite function f(g(x)), its derivative is f'(g(x)) * g'(x). In our case, the outer function is ln(u) and the inner function is u = x^4 - 3x - 1.
The derivative of ln(u) with respect to u is 1/u, and the derivative of u = x^4 - 3x - 1 with respect to x is 4x^3 - 3. Applying the chain rule, we get:
f'(x) = (1 / ln(9)) * (1 / (x^4 - 3x - 1)) * (4x^3 - 3)
Simplifying this expression, we have:
f'(x) = (4x^3 - 3) / ((x^4 - 3x - 1) * ln(9))
This is the derivative of the given function. Each step in this process is critical, from recognizing the applicability of the chain rule to accurately computing the derivatives of the inner and outer functions. The final expression provides a clear formula for the rate of change of the original function at any point x, highlighting the power of calculus in analyzing functions. Understanding and applying these differentiation techniques is essential for solving a wide range of problems in mathematics, physics, engineering, and other disciplines.
Completing the Original Expression
Referring back to the original problem, we had the following expression to complete:
f'(x) = (x + ?) / ((x^4 - 3x - 1) * ln(?))
Comparing this with the derivative we found:
f'(x) = (4x^3 - 3) / ((x^4 - 3x - 1) * ln(9))
We can see that the numerator in the original expression should be 4x^3 - 3, and the missing value inside the logarithm in the denominator should be 9. Therefore, the completed expression is:
f'(x) = (4x^3 - 3) / ((x^4 - 3x - 1) * ln(9))
This completion reinforces the importance of accurate differentiation and comparison. It also emphasizes the need for careful algebraic manipulation to match the derived result with the given format. This step not only completes the problem but also validates the correctness of our differentiation process. The final form of the derivative is now consistent with the expected structure, providing a clear and concise answer. This methodical approach to problem-solving is crucial in mathematics, where precision and accuracy are paramount.
Conclusion
In this exercise, we successfully rewrote and differentiated the function f(x) = log9(x^4 - 3x - 1). We began by applying the change of base formula to convert the logarithm to a natural logarithm, which is more convenient for differentiation. Then, we used the chain rule to find the derivative, resulting in f'(x) = (4x^3 - 3) / ((x^4 - 3x - 1) * ln(9)). Finally, we completed the original expression by identifying the missing terms, ensuring our solution aligned with the problem's requirements. This process highlights the importance of understanding logarithmic properties and differentiation rules, as well as the ability to apply them methodically. Mastery of these concepts is fundamental for further studies in calculus and its applications in various fields. The ability to rewrite functions before differentiation often simplifies the process and reduces the likelihood of errors, making it a valuable technique in mathematical problem-solving.
