Question

find the derivative of ( f(x) ).

( f(x)=e^{8 x}+e^{x} )

( f^{prime}(x)= )

Explanation:

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

Use the chain rule \((e^{u})^\prime=e^{u}\cdot u^\prime\). Let \(u = 8x\), then \(u^\prime=8\). So \((e^{8x})^\prime=e^{8x}\cdot8 = 8e^{8x}\).

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

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

Step3: Sum the derivatives

Since \(f(x)=e^{8x}+e^{x}\), then \(f^\prime(x)=(e^{8x})^\prime+(e^{x})^\prime\).

Answer:

\(8e^{8x}+e^{x}\)