Answer:
Slope of the Curve: [tex]f'(x)=\frac{-6}{x^2}[/tex]
Equation of Tangent Line: y + 3 = -3/2(x + 2)
General Formulas and Concepts:
Pre-Algebra
- Order of Operations: BPEMDAS
Algebra I
Point-Slope Form: y - y₁ = m(x - x₁)
- x₁ - x coordinate
- y₁ - y coordinate
- m - slope
Calculus
The definition of a derivative is the slope of the tangent line.
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
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
[tex]f(x)=\frac{6}{x}[/tex]
Step 2: Take Derivative
- Quotient Rule: [tex]f'(x)=\frac{0(x)-6(1)}{x^2}[/tex]
- Multiply: [tex]f'(x)=\frac{0-6}{x^2}[/tex]
- Subtract: [tex]f'(x)=\frac{-6}{x^2}[/tex]
Step 3: Find Instantaneous Derivative
- Substitute in x: [tex]f'(x)=\frac{-6}{(-2)^2}[/tex]
- Exponents: [tex]f'(x)=\frac{-6}{4}[/tex]
- Simplify: [tex]f'(x)=\frac{-3}{2}[/tex]
This value shows the slope of the tangent line at the exact value of x = 2.
- Substitute: y + 3 = -3/2(x + 2)
