Question

attempt 1: 10 attempts remaining. evaluate $\frac{d}{dt}(5e^{- 12t})$. $\frac{d}{dt}(5e^{-12t})=$

Explanation:

Step1: Use constant - multiple rule of differentiation

The constant - multiple rule states that if \(y = cf(x)\), then \(y^\prime=c\cdot f^\prime(x)\), where \(c = 5\) and \(f(t)=e^{-12t}\). So \(\frac{d}{dt}(5e^{-12t})=5\frac{d}{dt}(e^{-12t})\).

Step2: Use chain - rule for \(e^{-12t}\)

The chain - rule for \(y = e^{u}\) where \(u=-12t\) is \(\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}\). The derivative of \(y = e^{u}\) with respect to \(u\) is \(e^{u}\), and the derivative of \(u=-12t\) with respect to \(t\) is \(- 12\). So \(\frac{d}{dt}(e^{-12t})=e^{-12t}\cdot(-12)\).

Step3: Combine results

Substitute \(\frac{d}{dt}(e^{-12t})=-12e^{-12t}\) into \(5\frac{d}{dt}(e^{-12t})\). We get \(5\times(-12)e^{-12t}=-60e^{-12t}\).

Answer:

\(-60e^{-12t}\)