Question

Let y = 5e5z

Answer 1

Answer:

$\displaystyle dy = 25e^{5x}dx\\dy = 3.27 \cdot 10^7$

General Formulas and Concepts:

Math

• Rounding
• Euler's Number e - 2.71828

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Calculus

Derivatives

Derivative Notation

Differentials

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

eˣ Derivative: $\displaystyle \frac{dy}{dx}[e^u] = u'e^u$

Step-by-step explanation:

Part A

Step 1: Define

$\displaystyle y = 5e^{5x}$

Step 2: Differentiate

1. [Function] eˣ Derivative:                                                                                 $\displaystyle \frac{dy}{dx} = \frac{dy}{dx}[5x] \cdot 5e^{5x}$
2. [Derivative] Basic Power Rule:                                                                      $\displaystyle \frac{dy}{dx} = 5x^{1 - 1} \cdot 5e^{5x}$
3. [Derivative] Simplify:                                                                                       $\displaystyle \frac{dy}{dx} = 5 \cdot 5e^{5x}$
4. [Derivative] Multiply:                                                                                       $\displaystyle \frac{dy}{dx} = 25e^{5x}$
5. [Derivative] [Multiplication Property of Equality] Isolate dy:                        $\displaystyle dy = 25e^{5x}dx$

Part B

Step 1: Define

[Differential] $\displaystyle dy = 25e^{5x}dx$

[Given] x = 3, dx = 0.4

Step 2: Evaluate

1. Substitute in variables [Differential]:                                                             $\displaystyle dy = 25e^{5(3)}(0.4)$
2. [Differential] [Exponents] Multiply:                                                                $\displaystyle dy = 25e^{15}(0.4)$
3. [Differential] Evaluate exponents:                                                                 $\displaystyle dy = 25(3.26902 \cdot 10^6)(0.4)$
4. [Differential] Multiply:                                                                                     $\displaystyle dy = (8.17254 \cdot 10^7)(0.4)$
5. [Differential] Multiply:                                                                                     $\displaystyle dy = 3.26902 \cdot 10^7$
6. [Differential] Round:                                                                                       $\displaystyle dy = 3.27 \cdot 10^7$

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Differentials

Book: College Calculus 10e