Question

find the derivative of ( f(x) ).
( f(x)=2^{x}+3^{x} )
( f^{prime}(x)= )

Explanation:

Step1: Apply the sum rule of derivatives

The sum rule states that if \(f(x)=u(x)+v(x)\), then \(f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)\). Here, \(u(x) = 2^{x}\) and \(v(x)=3^{x}\), so \(f^{\prime}(x)=\frac{d}{dx}(2^{x})+\frac{d}{dx}(3^{x})\).

Step2: Use the formula for the derivative of \(a^{x}\)

The formula for the derivative of \(a^{x}\) is \(\frac{d}{dx}(a^{x})=a^{x}\ln(a)\).
For \(u(x) = 2^{x}\), \(u^{\prime}(x)=2^{x}\ln(2)\).
For \(v(x)=3^{x}\), \(v^{\prime}(x)=3^{x}\ln(3)\).

Answer:

\(2^{x}\ln(2)+3^{x}\ln(3)\)