The derivative of `log_(10)x` with respect to `x^(2)`, is

Options

A. `(1)/(2x^(2))log_(e)10`
B. `(2)/(x^(2))log_(10)e`
C. `(1)/(2x^(2))log_(10)^( e )`
D. none of these

Explanation

To find the derivative of \( \log_{10} x \) with respect to \( x^2 \), we can use the chain rule. Here’s the step-by-step solution:

### Step 1: Define the functions
Let:
- \( u = \log_{10} x \)
- \( v = x^2 \)

We want to find \( \frac{du}{dv} \).

### Step 2: Use the chain rule
Using the chain rule, we have:
\[ \frac{du}{dv} = \frac{du}{dx} \cdot \frac{dx}{dv} \]

### Step 3: Calculate \( \frac{du}{dx} \)
To find \( \frac{du}{dx} \):
\[ u = \log_{10} x = \frac{\ln x}{\ln 10} \]
Thus,
\[ \frac{du}{dx} = \frac{1}{x \ln 10} \]

### Step 4: Calculate \( \frac{dv}{dx} \)
To find \( \frac{dv}{dx} \):
\[ v = x^2 \]
Thus,
\[ \frac{dv}{dx} = 2x \]

### Step 5: Calculate \( \frac{dx}{dv} \)
Now, we need \( \frac{dx}{dv} \):
\[ \frac{dx}{dv} = \frac{1}{\frac{dv}{dx}} = \frac{1}{2x} \]

### Step 6: Combine the derivatives
Now substitute \( \frac{du}{dx} \) and \( \frac{dx}{dv} \) into the chain rule:
\[ \frac{du}{dv} = \frac{du}{dx} \cdot \frac{dx}{dv} = \left(\frac{1}{x \ln 10}\right) \cdot \left(\frac{1}{2x}\right) \]

### Step 7: Simplify the expression
This simplifies to:
\[ \frac{du}{dv} = \frac{1}{2x^2 \ln 10} \]

### Final Answer
Thus, the derivative of \( \log_{10} x \) with respect to \( x^2 \) is:
\[ \frac{1}{2x^2 \ln 10} \]

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