Mathematics, 22.11.2019 02:31 Queey

How do you use the limit definition of the derivative to find the derivative of f(x)=1/x?

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Answer from: maddie0533

$\displaystyle f'(x) = \frac{-1}{x^2}$

General Formulas and Concepts:

Algebra I

Terms/Coefficients

Expanding/Factoring

Functions

Function Notation

Calculus

Limits

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

Differentiation

DerivativesDerivative NotationDefinition of a Derivative: $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle f(x) = \frac{1}{x}$

Step 2: Differentiate

Substitute in function [Definition of a Derivative]: $\displaystyle f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h}$ Rewrite: $\displaystyle f'(x) = \lim_{h \to 0} \frac{x - (x + h)}{hx(x + h)}$ Expand: $\displaystyle f'(x) = \lim_{h \to 0} \frac{x - x - h}{hx(x + h)}$ Combine like terms: $\displaystyle f'(x) = \lim_{h \to 0} \frac{-h}{hx(x + h)}$ Simplify: $\displaystyle f'(x) = \lim_{h \to 0} \frac{-1}{x(x + h)}$ Evaluate limit [Limit Rule - Variable Direct Substitution]: $\displaystyle f'(x) = \frac{-1}{x(x + 0)}$ Simplify: $\displaystyle f'(x) = \frac{-1}{x^2}$

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

Unit: Differentiation

Answer from: Quest

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a pyramid is a three-dimensional object with a base that is a polygon and triangular faces that meet at one vertex.

" the net of a triangular pyramid consists of four triangles. the base here we are given of the pyramid is a equilateral triangle, and the lateral faces are also triangles "

Answer from: Quest

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