Question

find the derivative of ( f(x) ).

( f(x)=e^{4 x}+e^{x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(e^{4x}\)

Use the chain rule \((e^{u})^\prime=e^{u}\cdot u^\prime\). Let \(u = 4x\), then \(u^\prime=4\). So \((e^{4x})^\prime=e^{4x}\cdot4 = 4e^{4x}\)

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

For \(y = e^{x}\), its derivative is \(e^{x}\) according to the formula \((e^{x})^\prime=e^{x}\)

Step3: Sum the derivatives

Since \(f(x)=e^{4x}+e^{x}\), by the sum rule \((u + v)^\prime=u^\prime+v^\prime\), we have \(f^\prime(x)=(e^{4x})^\prime+(e^{x})^\prime\)

Answer:

\(4e^{4x}+e^{x}\)