find the derivative of ( f(x) ).\n\n( f(x)=e^{-3 x}-4^{x} )\n\n( f^{prime}(x)= )

Explanation:

Step1: Differentiate (e^{-3x})

Using the chain rule ((e^{u})^\prime=e^{u}\cdot u^\prime), where (u = - 3x) and (u^\prime=-3). So ((e^{-3x})^\prime=e^{-3x}\cdot(-3)=-3e^{-3x})

Step2: Differentiate (4^{x})

Using the formula ((a^{x})^\prime=a^{x}\ln a), where (a = 4). So ((4^{x})^\prime=4^{x}\ln4)

Step3: Find (f^\prime(x))

Since (f(x)=e^{-3x}-4^{x}), then (f^\prime(x)=(e^{-3x})^\prime-(4^{x})^\prime) Substitute the derivatives from Step1 and Step2: (f^\prime(x)=-3e^{-3x}-4^{x}\ln4)

Answer:

(f^\prime(x)=-3e^{-3x}-4^{x}\ln4)