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?

Answers: 3

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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

answer: 414,377,008

step-by-step explanation:

just multiply it

Answer from: Quest

step-by-step explanation:

she should of added 9^2 and 12^2. that is because it is a^2+b^2=c^2. c^2 is the hypotenuse and should always be bigger than the other two numbers. which is why she should of added 9 squared and 12 squared together instead of 15 squared.

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