Question

find the derivative of ( f(x) ).

( f(x)=8 e^{-x}+5 cdot 4^{x} )

( f^{prime}(x)= )

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

Answer:

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