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