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

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!