Question

find the derivative of ( f(x) ).

( f(x)=e^{x}+e^{3 x} )
( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(e^x\)

The derivative of \(e^x\) with respect to \(x\) is \(e^x\).

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

Using the chain rule \((e^{u})^\prime=e^{u}\cdot u^\prime\), where \(u = 3x\) and \(u^\prime=3\). So the derivative of \(e^{3x}\) is \(e^{3x}\cdot3 = 3e^{3x}\).

Step3: Sum the derivatives

\(f^\prime(x)=(e^x)^\prime+(e^{3x})^\prime\)

Answer:

\(e^x + 3e^{3x}\)