find the derivative of ( f(x) ).\n( f(x)=e^{-5 x}+9^{x} )\n( f^{prime}(x)= )

Explanation:

Step1: Differentiate (e^{-5x})

Use the chain rule ((e^{u})^\prime=e^{u}\cdot u^\prime). Let (u = - 5x), then (u^\prime=-5). So ((e^{-5x})^\prime=e^{-5x}\cdot(-5)=-5e^{-5x})

Step2: Differentiate (9^{x})

Use the formula ((a^{x})^\prime=a^{x}\ln a). Here (a = 9), so ((9^{x})^\prime=9^{x}\ln9)

Step3: Combine the derivatives

Since (f(x)=e^{-5x}+9^{x}), then (f^\prime(x)=(e^{-5x})^\prime+(9^{x})^\prime)

Answer:

(-5e^{-5x}+9^{x}\ln9)