Question

find the derivative of ( f(x) ).

( f(x)=e^{6 x}+e^{x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(e^{6x}\)

Use the chain rule \((e^{u})^\prime=e^{u}\cdot u^\prime\). Let \(u = 6x\), then \(u^\prime=6\). So \((e^{6x})^\prime=e^{6x}\cdot6 = 6e^{6x}\).

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

The derivative of \(e^{x}\) with respect to \(x\) is \(e^{x}\) (since for \(y = e^{x}\), \(y^\prime=e^{x}\) as the derivative of \(e^{u}\) with \(u=x\) and \(u^\prime = 1\), so \((e^{x})^\prime=e^{x}\cdot1=e^{x}\)).

Step3: Sum the derivatives

By the sum rule \((f(x)+g(x))^\prime=f^\prime(x)+g^\prime(x)\), if \(f(x)=e^{6x}\) and \(g(x)=e^{x}\), then \(f^\prime(x)+g^\prime(x)=6e^{6x}+e^{x}\).

Answer:

\(6e^{6x}+e^{x}\)