Question

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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Answer 1

`\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 (dy)/(dx)[e^u] = u'e^u`

Explanation:

Part A

Step 1: Define

`\displaystyle y = 5e^(5x)`

Step 2: Differentiate

1. [Function] eˣ Derivative:
   `\displaystyle (dy)/(dx) = (dy)/(dx)[5x] \cdot 5e^(5x)`
2. [Derivative] Basic Power Rule:
   `\displaystyle (dy)/(dx) = 5x^(1 - 1) \cdot 5e^(5x)`
3. [Derivative] Simplify:
   `\displaystyle (dy)/(dx) = 5 \cdot 5e^(5x)`
4. [Derivative] Multiply:
   `\displaystyle (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