Differentiate the following w.r.t. x :
`e^(mlogx)`

Answer

To differentiate the function \( y = e^{m \log x} \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function using properties of logarithms
We start with the function:
\[ y = e^{m \log x} \]
Using the property of exponents and logarithms, we can rewrite this as:
\[ y = x^m \]
This is because \( e^{\log a} = a \) and \( e^{m \log x} = (e^{\log x})^m = x^m \).

### Step 2: Differentiate the rewritten function
Now, we differentiate \( y = x^m \) with respect to \( x \):
\[ \frac{dy}{dx} = m x^{m-1} \]

### Step 3: Substitute back to original form (if necessary)
Since we have expressed \( y \) in terms of \( x \), we can also express the derivative in terms of the original function:
\[ \frac{dy}{dx} = m e^{m \log x} \cdot \frac{1}{x} \]
This is derived from the chain rule, where we recognize that \( e^{m \log x} = y \).

### Final Answer
Thus, the derivative of \( y = e^{m \log x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{m e^{m \log x}}{x} = \frac{m x^m}{x} = m x^{m-1} \]