Question

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Answer 1

Answer:

$\displaystyle f'(x) = \frac{4}{x^2}$

General Formulas and Concepts:

Calculus

Limits

• Limit Rule [Variable Direct Substitution]:                                                    $\displaystyle \lim_{x \to c} x = c$

Differentiation

• Derivatives
• Derivative Notation

The definition of a derivative is the slope of the tangent line:                             $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Step-by-step explanation:

Step 1: Define

Identify.

$\displaystyle f(x) = -\frac{4}{x}$

Step 2: Differentiate

1. [Function] Substitute in x:                                                                            $\displaystyle f(x + h) = -\frac{4}{x + h}$
2. Substitute in functions [Definition of a Derivative]:                                   $\displaystyle f'(x) = \lim_{h \to 0} \frac{-\frac{4}{x + h} - \big( -\frac{4}{x} \big)}{h}$
3. Simplify:                                                                                                        $\displaystyle f'(x) = \lim_{h \to 0} \frac{4}{x(x+ h)}$
4. Evaluate limit [Limit Rule - Variable Direct Substitution]:                          $\displaystyle f'(x) = \frac{4}{x(x+ 0)}$
5. Simplify:                                                                                                        $\displaystyle f'(x) = \frac{4}{x^2}$

∴ the derivative of the given function will be equal to 4 divided by x².

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Learn more about derivatives: brainly.com/question/25804880

Learn more about calculus: brainly.com/question/23558817

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

Unit: Differentiation