find the derivative of ( f(x) ).\n\n( f(x)=-5 cdot 9^{x}+3 e^{-x} )\n\n( f^{prime}(x)= )

Answer:

(-5\cdot9^{x}\ln9 - 3e^{-x})

Explanation:

Step1: Differentiate (-5\cdot9^{x})

Use the formula ((a^{x})^\prime=a^{x}\ln a). For (y = - 5\cdot9^{x}), by the constant - multiple rule ((cf(x))^\prime=c\cdot f^\prime(x)) (where (c=-5) and (f(x)=9^{x})), we have (y^\prime=-5\cdot9^{x}\ln9).

Step2: Differentiate (3e^{-x})

Use the formula ((e^{u(x)})^\prime = e^{u(x)}\cdot u^\prime(x)). Let (u(x)=-x), then (u^\prime(x)=-1). For (y = 3e^{-x}), by the constant - multiple rule ((cf(x))^\prime=c\cdot f^\prime(x)) (where (c = 3) and (f(x)=e^{-x})), we have (y^\prime=3\cdot e^{-x}\cdot(-1)=-3e^{-x}).

Step3: Combine the derivatives

By the sum rule ((f(x)+g(x))^\prime=f^\prime(x)+g^\prime(x)), if (f(x)=-5\cdot9^{x}) and (g(x)=3e^{-x}), then (f^\prime(x)+g^\prime(x)=-5\cdot9^{x}\ln9-3e^{-x}).