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Sagot :
Answer:
[tex]\displaystyle dy = 25e^{5x}dx\\dy = 3.27 \cdot 10^7[/tex]
General Formulas and Concepts:
Math
- Rounding
- Euler's Number e - 2.71828
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
- Left to Right
Calculus
Derivatives
Derivative Notation
Differentials
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
eˣ Derivative: [tex]\displaystyle \frac{dy}{dx}[e^u] = u'e^u[/tex]
Step-by-step explanation:
Part A
Step 1: Define
[tex]\displaystyle y = 5e^{5x}[/tex]
Step 2: Differentiate
- [Function] eˣ Derivative: [tex]\displaystyle \frac{dy}{dx} = \frac{dy}{dx}[5x] \cdot 5e^{5x}[/tex]
- [Derivative] Basic Power Rule: [tex]\displaystyle \frac{dy}{dx} = 5x^{1 - 1} \cdot 5e^{5x}[/tex]
- [Derivative] Simplify: [tex]\displaystyle \frac{dy}{dx} = 5 \cdot 5e^{5x}[/tex]
- [Derivative] Multiply: [tex]\displaystyle \frac{dy}{dx} = 25e^{5x}[/tex]
- [Derivative] [Multiplication Property of Equality] Isolate dy: [tex]\displaystyle dy = 25e^{5x}dx[/tex]
Part B
Step 1: Define
[Differential] [tex]\displaystyle dy = 25e^{5x}dx[/tex]
[Given] x = 3, dx = 0.4
Step 2: Evaluate
- Substitute in variables [Differential]: [tex]\displaystyle dy = 25e^{5(3)}(0.4)[/tex]
- [Differential] [Exponents] Multiply: [tex]\displaystyle dy = 25e^{15}(0.4)[/tex]
- [Differential] Evaluate exponents: [tex]\displaystyle dy = 25(3.26902 \cdot 10^6)(0.4)[/tex]
- [Differential] Multiply: [tex]\displaystyle dy = (8.17254 \cdot 10^7)(0.4)[/tex]
- [Differential] Multiply: [tex]\displaystyle dy = 3.26902 \cdot 10^7[/tex]
- [Differential] Round: [tex]\displaystyle dy = 3.27 \cdot 10^7[/tex]
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Differentials
Book: College Calculus 10e
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