Answer:
[tex]\displaystyle f'(x) = \frac{4}{x^2}[/tex]
General Formulas and Concepts:
Calculus
Limits
- Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]
Differentiation
- Derivatives
- Derivative Notation
The definition of a derivative is the slope of the tangent line: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]
Step-by-step explanation:
Step 1: Define
Identify.
[tex]\displaystyle f(x) = -\frac{4}{x}[/tex]
Step 2: Differentiate
- [Function] Substitute in x: [tex]\displaystyle f(x + h) = -\frac{4}{x + h}[/tex]
- Substitute in functions [Definition of a Derivative]: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{-\frac{4}{x + h} - \big( -\frac{4}{x} \big)}{h}[/tex]
- Simplify: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{4}{x(x+ h)}[/tex]
- Evaluate limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle f'(x) = \frac{4}{x(x+ 0)}[/tex]
- Simplify: [tex]\displaystyle f'(x) = \frac{4}{x^2}[/tex]
∴ the derivative of the given function will be equal to 4 divided by x².
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Learn more about derivatives: https://brainly.com/question/25804880
Learn more about calculus: https://brainly.com/question/23558817
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
