If `y=(1)/(log_(10)x)`, then what is `(dy)/(dx)` equal to?

Options

A. x
B. `x log_(e)10`
C. `-((log_(x)10)^(2)(log_(10)e))/(x)`
D. `xlog_(10)e`

Explanation

To find the derivative of the function \( y = \frac{1}{\log_{10} x} \), we will use the chain rule and the quotient rule. Here’s the step-by-step solution:

### Step 1: Rewrite the Function
We can rewrite the function \( y \) as:
\[ y = (\log_{10} x)^{-1} \]

### Step 2: Differentiate Using the Chain Rule
Using the chain rule, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -1 \cdot (\log_{10} x)^{-2} \cdot \frac{d}{dx}(\log_{10} x) \]

### Step 3: Differentiate \( \log_{10} x \)
To differentiate \( \log_{10} x \), we use the change of base formula:
\[ \log_{10} x = \frac{\ln x}{\ln 10} \]
Thus, the derivative is:
\[ \frac{d}{dx}(\log_{10} x) = \frac{1}{\ln 10} \cdot \frac{1}{x} = \frac{1}{x \ln 10} \]

### Step 4: Substitute Back into the Derivative
Now we substitute this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -(\log_{10} x)^{-2} \cdot \frac{1}{x \ln 10} \]

### Step 5: Simplify the Expression
This can be simplified to:
\[ \frac{dy}{dx} = -\frac{1}{x \ln 10 \cdot (\log_{10} x)^2} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\frac{1}{x \ln 10 \cdot (\log_{10} x)^2} \]

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