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