If `y=log_(10)x+log_(x)10+log_(x)x+log_(10)10` then what is `((dy)/(dx))_(x=10)` equal to?

Options

A. `1/(x log_(e)10)-(log_(e)10)/(x(log_(e)x)^(2))`
B. `1/(xlog_(e)10)-1/(xlog_(10)e)`
C. `1/(x log_(e)10)-1/(log_(e)x)^(2)`
D. None of these

Answer

To solve the problem step by step, we start with the given expression for \( y \):

### Step 1: Write down the expression for \( y \)
\[ y = \log_{10} x + \log_{x} 10 + \log_{x} x + \log_{10} 10 \]

### Step 2: Simplify the expression
Using the change of base formula for logarithms, we can rewrite \( \log_{x} 10 \) and \( \log_{x} x \):
\[ \log_{x} 10 = \frac{\log_{10} 10}{\log_{10} x} = \frac{1}{\log_{10} x} \]
\[ \log_{x} x = 1 \]
\[ \log_{10} 10 = 1 \]
Substituting these into the expression for \( y \):
\[ y = \log_{10} x + \frac{1}{\log_{10} x} + 1 + 1 \]
\[ y = \log_{10} x + \frac{1}{\log_{10} x} + 2 \]

### Step 3: Differentiate \( y \) with respect to \( x \)
Now, we differentiate \( y \):
\[ \frac{dy}{dx} = \frac{d}{dx} \left( \log_{10} x \right) + \frac{d}{dx} \left( \frac{1}{\log_{10} x} \right) + 0 \]
The derivative of \( \log_{10} x \) is:
\[ \frac{d}{dx} \left( \log_{10} x \right) = \frac{1}{x \ln 10} \]
For the second term, we use the quotient rule:
\[ \frac{d}{dx} \left( \frac{1}{\log_{10} x} \right) = -\frac{1}{(\log_{10} x)^2} \cdot \frac{d}{dx} \left( \log_{10} x \right) = -\frac{1}{(\log_{10} x)^2} \cdot \frac{1}{x \ln 10} \]
Combining these results, we have:
\[ \frac{dy}{dx} = \frac{1}{x \ln 10} - \frac{1}{x (\log_{10} x)^2 \ln 10} \]

### Step 4: Factor out common terms
Factoring out \( \frac{1}{x \ln 10} \):
\[ \frac{dy}{dx} = \frac{1}{x \ln 10} \left( 1 - \frac{1}{\log_{10} x^2} \right) \]

### Step 5: Evaluate at \( x = 10 \)
Now, we evaluate \( \frac{dy}{dx} \) at \( x = 10 \):
\[ \log_{10} 10 = 1 \]
Substituting \( x = 10 \):
\[ \frac{dy}{dx} \bigg|_{x=10} = \frac{1}{10 \ln 10} \left( 1 - \frac{1}{1^2} \right) = \frac{1}{10 \ln 10} (1 - 1) = \frac{1}{10 \ln 10} \cdot 0 = 0 \]

### Final Answer
Thus, the value of \( \frac{dy}{dx} \) at \( x = 10 \) is:
\[ \frac{dy}{dx} \bigg|_{x=10} = 0 \]