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

Options

A. x
B. `xlog_(e)10`
C. `-((log_(x)10)^(2)(log10^(e)))/(x)`
D. `xlog_(10)e`

Answer

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

### Step 1: Rewrite the function
We start with the function:
\[ y = \frac{1}{\log_{10} x} \]
This can be rewritten using the property of logarithms:
\[ y = (\log_{10} x)^{-1} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \), we apply the chain rule. The derivative of \( u^{-1} \) is \( -u^{-2} \cdot \frac{du}{dx} \), where \( u = \log_{10} x \). Thus,
\[ \frac{dy}{dx} = -(\log_{10} x)^{-2} \cdot \frac{d}{dx}(\log_{10} x) \]

### Step 3: Differentiate \( \log_{10} x \)
Now we need to find \( \frac{d}{dx}(\log_{10} x) \). Using the change of base formula, we have:
\[ \log_{10} x = \frac{\ln x}{\ln 10} \]
Thus,
\[ \frac{d}{dx}(\log_{10} x) = \frac{1}{\ln 10} \cdot \frac{1}{x} = \frac{1}{x \ln 10} \]

### 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 simplifies to:
\[ \frac{dy}{dx} = -\frac{1}{x \ln 10 (\log_{10} x)^2} \]

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

---