Answer:
\(\displaystyle \frac{ds}{dt} = 12e^{3t}\)
General Formulas and Concepts:
Algebra I
- Functions
- Function Notation
Calculus
Derivatives
Derivative Notation
Derivative Property [Multiplied Constant]: \(\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)\)
Derivative Property [Addition/Subtraction]: \(\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]\)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Chain Rule]: \(\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)\)
eˣ Derivative: \(\displaystyle \frac{d}{dx} [e^u]=e^u \cdot u'\)
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle s = 4e^{3t} - e^{-2.5}\)
Step 2: Differentiate
- eˣ Derivative: \(\displaystyle \frac{ds}{dt} = 4e^{3t} \cdot \frac{d}{dt}[3t] - \frac{d}{dt}[e^{-2.5}]\)
- Basic Power Rule: \(\displaystyle \frac{ds}{dt} = 4e^{3t} \cdot 3t^{1 - 1} - 0\)
- Simplify: \(\displaystyle \frac{ds}{dt} = 12e^{3t}\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
