Question

find the derivative of ( f(x) ).

( f(x)=e^{5 x}-e^{4 x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(e^{5x}\)

Use the chain rule \((e^{u})^\prime = e^{u}\cdot u^\prime\). Let \(u = 5x\), then \(u^\prime=5\). So \((e^{5x})^\prime=e^{5x}\cdot5 = 5e^{5x}\)

Step2: Differentiate \(e^{4x}\)

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

Step3: Apply the difference rule

Since \(f(x)=e^{5x}-e^{4x}\), by \((u - v)^\prime=u^\prime - v^\prime\), we have \(f^\prime(x)=(e^{5x})^\prime-(e^{4x})^\prime\)
Substitute the results from Step1 and Step2: \(f^\prime(x)=5e^{5x}-4e^{4x}\)

Answer:

\(5e^{5x}-4e^{4x}\)