Question

find the derivative of ( f(x) ).

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

( 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:

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