Question

If $y = \log_{10} x + \log_x 10 + \log_x x + \log_{10} 10$, then $\frac{dy}{dx} = $

• A. $\frac{1}{x \log_e 10} + \frac{1}{x \log_{10} e}$
• B. $\frac{1}{x \log_e 10} + \frac{\log_e 10}{x (\log_{10} e)^2}$
• C. $\frac{1}{x \log_e 10} - \frac{1}{x \log_{10} e}$
• D. $\frac{1}{x \log_e 10} - \frac{\log_e 10}{x (\log_e x)^2}$

Hint:

Remember that terms like $\log_x x$ and $\log_{k} k$ are just constants equal to 1. Identify them early and drop them from your calculations immediately, since the derivative of any constant is zero!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a function $y$ containing multiple logarithmic expressions with different bases. We need to find its first derivative $\frac{dy}{dx}$ with respect to $x$.

Step 2: Key Formula or Approach:
Before differentiating, we must convert all logarithms to the natural base $e$ using the change-of-base formula: $$\log_b a = \frac{\log_e a}{\log_e b}$$ Note that $\log_x x = 1$ and $\log_{10} 10 = 1$ are constant terms whose derivatives are zero. We will use standard derivative rules: $$\frac{d}{dx}(\log_e x) = \frac{1}{x} \quad \text{and} \quad \frac{d}{dx}\left(\frac{1}{f(x)}\right) = -\frac{f'(x)}{[f(x)]^2}$$

Step 3: Detailed Explanation:
Let's simplify the given expression for $y$ using the change-of-base rules: $$y = \frac{\log_e x}{\log_e 10} + \frac{\log_e 10}{\log_e x} + 1 + 1$$ $$y = \left(\frac{1}{\log_e 10}\right) \log_e x + (\log_e 10) (\log_e x)^{-1} + 2$$ Now, differentiate each term with respect to $x$: $$\frac{dy}{dx} = \left(\frac{1}{\log_e 10}\right) \cdot \frac{1}{x} + (\log_e 10) \cdot \left[ -1 \cdot (\log_e x)^{-2} \cdot \frac{1}{x} \right] + 0$$ $$\frac{dy}{dx} = \frac{1}{x \log_e 10} - \frac{\log_e 10}{x (\log_e x)^2}$$ This matches option (D).

Step 4: Final Answer:
The derivative of the function is $\frac{1}{x \log_e 10} - \frac{\log_e 10}{x (\log_e x)^2}$, which corresponds to option (D).