Differentiate `a^lnx` with respect to x

Answer

To differentiate \( a^{\ln x} \) with respect to \( x \), we can follow these steps:

### Step 1: Define the function
Let \( y = a^{\ln x} \).

### Step 2: Use the properties of exponents
We can rewrite \( y \) using the exponential function:
\[ y = e^{\ln(a^{\ln x})} = e^{\ln x \cdot \ln a} \]

### Step 3: Differentiate using the chain rule
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(e^{\ln x \cdot \ln a}) \]
Using the chain rule, we get:
\[ \frac{dy}{dx} = e^{\ln x \cdot \ln a} \cdot \frac{d}{dx}(\ln x \cdot \ln a) \]

### Step 4: Differentiate the inner function
Now we differentiate \( \ln x \cdot \ln a \):
\[ \frac{d}{dx}(\ln x \cdot \ln a) = \ln a \cdot \frac{d}{dx}(\ln x) = \ln a \cdot \frac{1}{x} \]

### Step 5: Substitute back into the derivative
Substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = e^{\ln x \cdot \ln a} \cdot \left(\ln a \cdot \frac{1}{x}\right) \]

### Step 6: Rewrite in terms of \( y \)
Since \( e^{\ln x \cdot \ln a} = a^{\ln x} \), we can write:
\[ \frac{dy}{dx} = a^{\ln x} \cdot \frac{\ln a}{x} \]

### Final Answer
Thus, the derivative of \( a^{\ln x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{a^{\ln x} \ln a}{x} \]

---