Question

evaluate (\frac{d}{dt}(3e^{-11t})). (\frac{d}{dt}(3e^{-11t}) = ) attempt 1: 10 attempts remaining. submit answer next item

Explanation:

Step1: Identify the function type

We have a function of the form \( y = 3e^{kt} \), where \( k=- 11 \). The derivative of \( e^{at} \) with respect to \( t \) is \( ae^{at} \) (by the chain rule, since the derivative of the exponent \( at \) with respect to \( t \) is \( a \), and the derivative of \( e^u \) with respect to \( u \) is \( e^u \), so by chain rule \( \frac{d}{dt}(e^{at})=ae^{at} \)). Also, the constant multiple rule states that \( \frac{d}{dt}(cf(t)) = c\frac{d}{dt}(f(t)) \) where \( c \) is a constant.

Step2: Apply the constant multiple and chain rule

For the function \( f(t)=3e^{- 11t} \), using the constant multiple rule with \( c = 3 \) and \( f(t)=e^{-11t} \), and then the chain rule on \( e^{-11t} \). The derivative of \( - 11t \) with respect to \( t \) is \( - 11 \), so the derivative of \( e^{-11t} \) with respect to \( t \) is \( - 11e^{-11t} \). Then, multiplying by the constant 3, we get:

\( \frac{d}{dt}(3e^{-11t})=3\times(- 11)e^{-11t} \)

Step3: Simplify the expression

Calculate \( 3\times(-11)=-33 \), so the derivative is \( - 33e^{-11t} \)

Answer:

\( -33e^{-11t} \)