find the derivative of ( f(x) ).\n\n( f(x)=e^{5 x}-e^{4 x} )\n\n( f^{prime}(x)= )
find the derivative of ( f(x) ).\n\n( f(x)=e^{5 x}-e^{4 x} )\n\n( f^{prime}(x)= )
Answer
Answer:
(5e^{5x}-4e^{4x})
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})