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 \( y = \log_{10} x \), we can follow these steps:

### Step 1: Rewrite the logarithm in terms of natural logarithm
We know that the logarithm with a different base can be expressed using the natural logarithm (base \( e \)). Using the change of base formula, we have:
\[ y = \log_{10} x = \frac{\ln x}{\ln 10} \]

### Step 2: Differentiate both sides with respect to \( x \)
Now, we differentiate both sides 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 factor it out:
\[ \frac{dy}{dx} = \frac{1}{\ln 10} \cdot \frac{d}{dx} (\ln x) \]

### Step 3: Differentiate \( \ln x \)
The derivative of \( \ln x \) is:
\[ \frac{d}{dx} (\ln x) = \frac{1}{x} \]
So, 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
Thus, we can write the final expression for the derivative as:
\[ \frac{dy}{dx} = \frac{1}{x \ln 10} \]

### Summary
The value of \( \frac{dy}{dx} \) when \( y = \log_{10} x \) is:
\[ \frac{dy}{dx} = \frac{1}{x \ln 10} \]

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