If `y = log_(10) x`, then the value of `(dy)/(dx)` is

Options

A. `1/x`
B. `x/(log_e 10)`
C. `(log_(10)e)/x`
D. `(log_e 10)/x`

Answer

To find the derivative of the function \( y = \log_{10} x \), we can follow these steps:

### Step 1: Rewrite the logarithm in terms of natural logarithm
Using the change of base formula for logarithms, we can express \( y \) as:
\[ y = \log_{10} x = \frac{\ln x}{\ln 10} \]
where \( \ln \) denotes the natural logarithm.

### Step 2: Differentiate with respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{\ln x}{\ln 10}\right) \]
Since \( \ln 10 \) is a constant, we can take it out of the derivative:
\[ \frac{dy}{dx} = \frac{1}{\ln 10} \cdot \frac{d}{dx}(\ln x) \]

### Step 3: Differentiate \( \ln x \)
The derivative of \( \ln x \) is:
\[ \frac{d}{dx}(\ln x) = \frac{1}{x} \]
Thus, substituting this back into our equation gives:
\[ \frac{dy}{dx} = \frac{1}{\ln 10} \cdot \frac{1}{x} \]

### Step 4: Final expression for the derivative
Therefore, we can express the derivative as:
\[ \frac{dy}{dx} = \frac{1}{x \ln 10} \]

### Conclusion
The final result for the derivative of \( y = \log_{10} x \) is:
\[ \frac{dy}{dx} = \frac{1}{x \ln 10} \]

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