find the derivative of ( f(x) ).\n\n( f(x)=9^{x}+e^{x} )\n\n( f^{prime}(x)= )

Explanation:

Step1: Derivative of (a^x)

The derivative of (a^x) is (a^x\ln a). For (y = 9^x), using the formula (\frac{d}{dx}(a^x)=a^x\ln a) (where (a = 9)), we have (\frac{d}{dx}(9^x)=9^x\ln 9).

Step2: Derivative of (e^x)

The derivative of (e^x) is (e^x), i.e., (\frac{d}{dx}(e^x)=e^x).

Step3: Sum - rule of derivatives

If (f(x)=u(x)+v(x)), then (f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x)). Here (u(x)=9^x) and (v(x)=e^x). So (f^{\prime}(x)=\frac{d}{dx}(9^x)+\frac{d}{dx}(e^x)).

Answer:

(9^x\ln 9 + e^x)