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Let y = 5e5z
A. Find the differential dy
25e53
dy
B. Use part A. to find dy when x = - 3 and dir = 0.4.
Round your answer to 2 decimal(s).
dy =
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Let Y 5e5z A Find The Differential Dy 25e53 Dy B Use Part A To Find Dy When X 3 And Dir 04 Round Your Answer To 2 Decimals Dy Submit Question class=

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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

  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: [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

  1. [Function] eˣ Derivative:                                                                                 [tex]\displaystyle \frac{dy}{dx} = \frac{dy}{dx}[5x] \cdot 5e^{5x}[/tex]
  2. [Derivative] Basic Power Rule:                                                                      [tex]\displaystyle \frac{dy}{dx} = 5x^{1 - 1} \cdot 5e^{5x}[/tex]
  3. [Derivative] Simplify:                                                                                       [tex]\displaystyle \frac{dy}{dx} = 5 \cdot 5e^{5x}[/tex]
  4. [Derivative] Multiply:                                                                                       [tex]\displaystyle \frac{dy}{dx} = 25e^{5x}[/tex]
  5. [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

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