Question

Implicit differentiation Please help

Answer 1

Hope this helps you with implicit differentiations

Answer 2

Answer:

$y''(-1) =8$

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

Equality Properties

Algebra I

• Factoring

Calculus

Implicit Differentiation

The derivative of a constant is equal to 0

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Product Rule: $\frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Chain Rule: $\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Quotient Rule: $\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

-xy - 2y = -4

Rate of change of the tangent line at point (-1, 4)

Step 2: Differentiate Pt. 1

Find 1st Derivative

1. Implicit Differentiation [Product Rule/Basic Power Rule]:                            $-y - xy' - 2y' = 0$
2. [Algebra] Isolate y' terms:                                                                               $-xy' - 2y' = y$
3. [Algebra] Factor y':                                                                                       $y'(-x - 2) = y$
4. [Algebra] Isolate y':                                                                                         $y' = \frac{y}{-x-2}$
5. [Algebra] Rewrite:                                                                                           $y' = \frac{-y}{x+2}$

Step 3: Find y

1. Define equation:                    $-xy - 2y = -4$
2. Factor y:                                 $y(-x - 2) = -4$
3. Isolate y:                                 $y = \frac{-4}{-x-2}$
4. Simplify:                                 $y = \frac{4}{x+2}$

Step 4: Rewrite 1st Derivative

1. [Algebra] Substitute in y:                                                                               $y' = \frac{-\frac{4}{x+2} }{x+2}$
2. [Algebra] Simplify:                                                                                         $y' = \frac{-4}{(x+2)^2}$

Step 5: Differentiate Pt. 2

Find 2nd Derivative

1. Differentiate [Quotient Rule/Basic Power Rule]:                                          $y'' = \frac{0(x+2)^2 - 8 \cdot 2(x + 2) \cdot 1}{[(x + 2)^2]^2}$
2. [Derivative] Simplify:                                                                                      $y'' = \frac{8}{(x+2)^3}$

Step 6: Find Slope at Given Point

1. [Algebra] Substitute in x:                                                                               $y''(-1) = \frac{8}{(-1+2)^3}$
2. [Algebra] Evaluate:                                                                                       $y''(-1) =8$