If `y = log_(10) x " then "(dy)/(dx)`= ?

Options

A. `(1)/(x)`
B. `(1)/(x) (log 10)`
C. `(1)/(x(log10))`
D. none of these

Explanation

To find the derivative of \( y = \log_{10} x \), we will use the change of base formula and the properties of logarithms. Here’s the step-by-step solution:

### Step 1: Rewrite the logarithm using the change of base formula
The change of base formula states that:
\[ \log_{a} b = \frac{\log_{e} b}{\log_{e} a} \]
Using this formula, we can rewrite \( y = \log_{10} x \) as:
\[ y = \frac{\log_{e} x}{\log_{e} 10} \]

### Step 2: Differentiate with respect to \( x \)
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{\log_{e} 10} \cdot \frac{d}{dx}(\log_{e} x) \]
We know that the derivative of \( \log_{e} x \) is \( \frac{1}{x} \). Therefore:
\[ \frac{dy}{dx} = \frac{1}{\log_{e} 10} \cdot \frac{1}{x} \]

### Step 3: Simplify the expression
Now we can simplify the expression:
\[ \frac{dy}{dx} = \frac{1}{x \log_{e} 10} \]

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