Find the derivative of f left parenthesis x right parenthesis equals ln open parentheses fraction numerator x to the power of 6 over denominator 2 x to the power of 5 minus 7 end fraction close parentheses.


f apostrophe left parenthesis x right parenthesis equals 6 ln left parenthesis x right parenthesis minus ln open parentheses 2 x to the power of 5 minus 7 close parentheses


f apostrophe left parenthesis x right parenthesis equals 6 over x minus fraction numerator 10 x to the power of 4 over denominator 2 x to the power of 5 minus 7 end fraction


f apostrophe left parenthesis x right parenthesis equals 6 over x minus fraction numerator 1 over denominator 2 x to the power of 5 minus 7 end fraction


f apostrophe left parenthesis x right parenthesis equals 6 over x plus fraction numerator 10 x to the power of 4 over denominator 2 x to the power of 5 minus 7 end fraction

To find the derivative of the function \( f(x) = \ln\left(\frac{x^6}{2x^5 - 7}\right) \), we will use the properties of logarithms and the chain rule.

First, simplify the logarithmic expression using logarithm properties:
\[ f(x) = \ln\left(\frac{x^6}{2x^5 - 7}\right) = \ln(x^6) - \ln(2x^5 - 7) \]

Now, differentiate term by term:

1. For \(\ln(x^6)\):
\[ \frac{d}{dx}[\ln(x^6)] = \frac{d}{dx}[6 \ln(x)] = 6 \cdot \frac{1}{x} = \frac{6}{x} \]

2. For \(\ln(2x^5 - 7)\):
\[ \frac{d}{dx}[\ln(2x^5 - 7)] = \frac{1}{2x^5 - 7} \cdot \frac{d}{dx}[2x^5 - 7] = \frac{1}{2x^5 - 7} \cdot 10x^4 = \frac{10x^4}{2x^5 - 7} \]

Combining these results, we get:
\[ f'(x) = \frac{6}{x} - \frac{10x^4}{2x^5 - 7} \]

Comparing with the given choices, the correct one is:
\[ f'(x) = \frac{6}{x} - \frac{10x^4}{2x^5 - 7} \]

Thus, the correct option is:
\[ f'(x) = \frac{6}{x} - \frac{10x^4}{2x^5 - 7} \]

Do you want more details or have any questions?

Here are 5 more questions you might want to ask:
1. How do you integrate the function \(f(x) = \ln\left(\frac{x^6}{2x^5 - 7}\right)\)?
2. What are the critical points of the function \(f(x) = \ln\left(\frac{x^6}{2x^5 - 7}\right)\)?
3. How do you find the second derivative of \(f(x) = \ln\left(\frac{x^6}{2x^5 - 7}\right)\)?
4. Can you explain the properties of logarithms used in differentiation?
5. How do you determine the concavity of the function \(f(x) = \ln\left(\frac{x^6}{2x^5 - 7}\right)\)?

Tip: Always simplify the logarithmic expressions using logarithm properties before differentiating to make the process easier.

Learn how to find the derivative of ln(x^6 / (2x^5 - 7)) using logarithm properties and the chain rule. Step-by-step explanation with examples. Suitable for advanced high school students and college-level calculus.

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