Derivative of `log_(10)x` with respect to ` x^(2)` is

Options

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

Explanation

To find the derivative of \( \log_{10} x \) with respect to \( x^2 \), we will follow these steps:

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

### Step 2: Find \( \frac{du}{dx} \)
Using the change of base formula for logarithms:
\[ \log_{10} x = \frac{\ln x}{\ln 10} \]
Now, differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{1}{\ln 10} \cdot \frac{1}{x} = \frac{1}{x \ln 10} \]

### Step 3: Find \( \frac{dv}{dx} \)
Now differentiate \( v \) with respect to \( x \):
\[ \frac{dv}{dx} = 2x \]

### Step 4: Use the chain rule to find \( \frac{du}{dv} \)
We need to find \( \frac{du}{dv} \):
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{\frac{1}{x \ln 10}}{2x} = \frac{1}{2x^2 \ln 10} \]

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

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