find the derivative of ( f(x) ).\n\n( f(x)=8 e^{-x}+5 cdot 4^{x} )\n\n( f^{prime}(x)= )

Answer:

(-8e^{-x}+5\cdot4^{x}\ln4)

Explanation:

Step1: Differentiate (8e^{-x})

Use the chain rule ((e^{u})^\prime = e^{u}\cdot u^\prime). Let (u=-x), then (u^\prime=-1). So ((8e^{-x})^\prime=8e^{-x}\cdot(-1)=-8e^{-x})

Step2: Differentiate (5\cdot4^{x})

Use the formula ((a^{x})^\prime=a^{x}\ln a). Here (a = 4), so ((5\cdot4^{x})^\prime=5\cdot4^{x}\ln4)

Step3: Combine the derivatives

By the sum rule ((f + g)^\prime=f^\prime+g^\prime), (f^\prime(x)=(8e^{-x})^\prime+(5\cdot4^{x})^\prime=-8e^{-x}+5\cdot4^{x}\ln4)