Question

find the derivative of the function.
y = 5e^{-x}+e^{4x}
\frac{dy}{dx}=\square

Explanation:

Step1: Apply sum - rule of derivatives

The derivative of a sum of functions \(y = u + v\) is \(\frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}\). Let \(u = 5e^{-x}\) and \(v=e^{4x}\). So \(\frac{dy}{dx}=\frac{d(5e^{-x})}{dx}+\frac{d(e^{4x})}{dx}\).

Step2: Find the derivative of \(u = 5e^{-x}\)

Using the constant - multiple rule \(\frac{d(cf(x))}{dx}=c\frac{df(x)}{dx}\) and the chain - rule \(\frac{d(e^{ax})}{dx}=ae^{ax}\), for \(u = 5e^{-x}\), we have \(\frac{d(5e^{-x})}{dx}=5\frac{d(e^{-x})}{dx}\). Since \(\frac{d(e^{-x})}{dx}=-e^{-x}\), then \(\frac{d(5e^{-x})}{dx}=5\times(-e^{-x})=- 5e^{-x}\).

Step3: Find the derivative of \(v = e^{4x}\)

Using the chain - rule \(\frac{d(e^{ax})}{dx}=ae^{ax}\), for \(v = e^{4x}\) with \(a = 4\), we get \(\frac{d(e^{4x})}{dx}=4e^{4x}\).

Step4: Combine the results

\(\frac{dy}{dx}=-5e^{-x}+4e^{4x}\).

Answer:

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