The derivative of `a^(x)` is :

Options

A. `a^(x)`
B. `a^(x)/loga`
C. `a^(x)loga`
D. None of these

Answer

To find the derivative of \( a^x \), we will follow these steps:

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

### Step 2: Take the natural logarithm of both sides
Taking the natural logarithm on both sides gives us:
\[ \ln y = \ln(a^x) \]

### Step 3: Simplify using logarithmic properties
Using the property of logarithms that states \( \ln(a^b) = b \ln a \), we can simplify:
\[ \ln y = x \ln a \]

### Step 4: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\ln y) = \frac{d}{dx}(x \ln a) \]

### Step 5: Apply the chain rule on the left side
Using the chain rule on the left side, we have:
\[ \frac{1}{y} \frac{dy}{dx} = \ln a \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = y \cdot \ln a \]

### Step 7: Substitute back for \( y \)
Since we defined \( y = a^x \), we substitute back:
\[ \frac{dy}{dx} = a^x \cdot \ln a \]

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

---