Answer:
[tex]y''(-1) =8[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Equality Properties
Algebra I
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: [tex]\frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Chain Rule: [tex]\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Quotient Rule: [tex]\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
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
- Implicit Differentiation [Product Rule/Basic Power Rule]: [tex]-y - xy' - 2y' = 0[/tex]
- [Algebra] Isolate y' terms: [tex]-xy' - 2y' = y[/tex]
- [Algebra] Factor y': [tex]y'(-x - 2) = y[/tex]
- [Algebra] Isolate y': [tex]y' = \frac{y}{-x-2}[/tex]
- [Algebra] Rewrite: [tex]y' = \frac{-y}{x+2}[/tex]
Step 3: Find y
- Define equation: [tex]-xy - 2y = -4[/tex]
- Factor y: [tex]y(-x - 2) = -4[/tex]
- Isolate y: [tex]y = \frac{-4}{-x-2}[/tex]
- Simplify: [tex]y = \frac{4}{x+2}[/tex]
Step 4: Rewrite 1st Derivative
- [Algebra] Substitute in y: [tex]y' = \frac{-\frac{4}{x+2} }{x+2}[/tex]
- [Algebra] Simplify: [tex]y' = \frac{-4}{(x+2)^2}[/tex]
Step 5: Differentiate Pt. 2
Find 2nd Derivative
- Differentiate [Quotient Rule/Basic Power Rule]: [tex]y'' = \frac{0(x+2)^2 - 8 \cdot 2(x + 2) \cdot 1}{[(x + 2)^2]^2}[/tex]
- [Derivative] Simplify: [tex]y'' = \frac{8}{(x+2)^3}[/tex]
Step 6: Find Slope at Given Point
- [Algebra] Substitute in x: [tex]y''(-1) = \frac{8}{(-1+2)^3}[/tex]
- [Algebra] Evaluate: [tex]y''(-1) =8[/tex]
