Question

find the derivative of ( f(x) ).

( f(x)=e^{8 x}+e^{7 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 \(e^{7x}\)

Use the chain rule \((e^{u})^\prime = e^{u}\cdot u^\prime\). Let \(u = 7x\), then \(u^\prime = 7\). So \((e^{7x})^\prime=e^{7x}\cdot7=7e^{7x}\)

Step3: Sum the derivatives

Since \(f(x)=e^{8x}+e^{7x}\), by the sum rule \((u + v)^\prime=u^\prime + v^\prime\), \(f^\prime(x)=(e^{8x})^\prime+(e^{7x})^\prime\)

Answer:

\(8e^{8x}+7e^{7x}\)