Differentiate w.r.t. as indicated :
`log_(10)x" w.r.t. "x^(2)`

Answer

To differentiate \( \log_{10} x \) with respect to \( x^2 \), we can follow these steps:

### Step 1: Use the Change of Variables
We recognize that we are differentiating \( \log_{10} x \) with respect to \( x^2 \). We can express this as:

\[ \frac{d}{dx^2} \log_{10} x \]

### Step 2: Apply the Chain Rule
Using the chain rule, we can rewrite the differentiation as:

\[ \frac{d}{dx^2} \log_{10} x = \frac{d}{dx} \log_{10} x \cdot \frac{dx}{dx^2} \]

### Step 3: Differentiate \( \log_{10} x \)
To differentiate \( \log_{10} x \) with respect to \( x \), we use the change of base formula:

\[ \log_{10} x = \frac{\ln x}{\ln 10} \]

Thus, the derivative is:

\[ \frac{d}{dx} \log_{10} x = \frac{1}{\ln 10} \cdot \frac{1}{x} = \frac{1}{x \ln 10} \]

### Step 4: Differentiate \( x^2 \)
Next, we differentiate \( x^2 \) with respect to \( x \):

\[ \frac{dx}{dx^2} = \frac{1}{2x} \]

### Step 5: Combine the Results
Now we can substitute back into our expression from Step 2:

\[ \frac{d}{dx^2} \log_{10} x = \frac{1}{x \ln 10} \cdot \frac{1}{2x} \]

This simplifies to:

\[ \frac{1}{2x^2 \ln 10} \]

### 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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