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

Explanation:

Step1: Differentiate (e^{-9x})

Use the chain rule. Let (u = -9x), then (\frac{d}{dx}(e^u)=e^u\cdot\frac{du}{dx}). Here (\frac{du}{dx}=-9), so (\frac{d}{dx}(e^{-9x})=-9e^{-9x}).

Step2: Differentiate (9^x)

Use the formula (\frac{d}{dx}(a^x)=a^x\ln a). For (a = 9), (\frac{d}{dx}(9^x)=9^x\ln 9).

Answer:

(-9e^{-9x}+9^x\ln 9)