$$ ( a ^ { x } ) ^ { \prime } $$

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To find the derivative of the function $$ f(x) = a^{x} $$ with respect to $$ x $$, where $$ a $$ is a constant, you can use the following approach:

### Using Exponential Differentiation

The derivative of an exponential function with a constant base is given by:

$$
\frac{d}{dx} a^{x} = a^{x} \ln(a)
$$

### Explanation

1. **Understanding the Function:**
   - $$ a^{x} $$ is an exponential function where the base $$ a $$ is a constant, and the exponent is the variable $$ x $$.

2. **Applying the Derivative Rule:**
   - The general rule for the derivative of $$ a^{x} $$ is derived from the definition of the exponential function and natural logarithms.
   - The derivative involves multiplying the original function by the natural logarithm of the base.

3. **Final Result:**
   - Therefore, the derivative $$ (a^{x})' $$ is:

$$
(a^{x})' = a^{x} \ln(a)
$$

### Example

If $$ a = 2 $$, then:

$$
\frac{d}{dx} 2^{x} = 2^{x} \ln(2)
$$

This means that the slope of the tangent line to the curve $$ y = 2^{x} $$ at any point $$ x $$ is $$ 2^{x} \ln(2) $$.

### Special Case: Natural Exponential Function

When the base $$ a $$ is Euler's number $$ e $$ (approximately 2.71828), the function becomes the natural exponential function:

$$
f(x) = e^{x}
$$

Its derivative simplifies to:

$$
\frac{d}{dx} e^{x} = e^{x}
$$

This is because $$ \ln(e) = 1 $$, so:

$$
\frac{d}{dx} e^{x} = e^{x} \cdot 1 = e^{x}
$$

### Summary

- **General Derivative:** $$ (a^{x})' = a^{x} \ln(a) $$
- **Natural Exponential Function:** $$ (e^{x})' = e^{x} $$

Feel free to ask if you have any more questions related to calculus or other topics!
The derivative of $$ a^{x} $$ with respect to $$ x $$ is $$ a^{x} \ln(a) $$.

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