Question

Find the derivative of

Answer 1

Answer:

$\displaystyle y'(1, \frac{3}{2}) = -3$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

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

• Left to Right

Algebra I

• Coordinates (x, y)
• Functions
• Function Notation
• Terms/Coefficients
• Exponential Rule [Rewrite]:                                                                              $\displaystyle b^{-m} = \frac{1}{b^m}$

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Step-by-step explanation:

Step 1: Define

$\displaystyle y = \frac{3}{2x^2}$

$\displaystyle \text{Point} \ (1, \frac{3}{2})$

Step 2: Differentiate

1. [Function] Rewrite [Exponential Rule - Rewrite]:                                            $\displaystyle y = \frac{3}{2}x^{-2}$
2. Basic Power Rule:                                                                                             $\displaystyle y' = -2 \cdot \frac{3}{2}x^{-2 - 1}$
3. Simplify:                                                                                                             $\displaystyle y' = -3x^{-3}$
4. Rewrite [Exponential Rule - Rewrite]:                                                              $\displaystyle y' = \frac{-3}{x^3}$

Step 3: Solve

1. Substitute in coordinate [Derivative]:                                                              $\displaystyle y'(1, \frac{3}{2}) = \frac{-3}{1^3}$
2. Evaluate exponents:                                                                                         $\displaystyle y'(1, \frac{3}{2}) = \frac{-3}{1}$
3. Divide:                                                                                                               $\displaystyle y'(1, \frac{3}{2}) = -3$

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

Unit: Derivatives

Book: College Calculus 10e