Question

find the derivative of $f(x)$. $f(x)=e^{-5x}+e^{4x}$ $f(x)=$ submit

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 \((e^{-5x})^\prime=e^{-5x}\cdot(-5)=-5e^{-5x}\).

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: Find \(f^\prime(x)\)

Since \(f(x)=e^{-5x}+e^{4x}\), then \(f^\prime(x)=(e^{-5x})^\prime+(e^{4x})^\prime\). Substitute the results from Step1 and Step2: \(f^\prime(x)=-5e^{-5x}+4e^{4x}\).

Answer:

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