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Finding the derivative by the limit process.

f(x)=x²+x-3
f'(x)=?​

User Lohardt
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1 Answer

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Answer:


\displaystyle f'(x) = 2x + 1

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

  • Derivatives
  • Derivative Notation
  • Definition of a Derivative:
    \displaystyle f'(x) = \lim_(h \to 0) (f(x + h) - f(x))/(h)

Explanation:

Step 1: Define

Identify


f(x) = x^2 + x - 3

Step 2: Differentiate

  1. Substitute in function [Definition of a Derivative]:
    \displaystyle f'(x) = \lim_(h \to 0) ([(x + h)^2 + (x + h) - 3] - (x^2 + x - 3))/(h)
  2. Expand:
    \displaystyle f'(x) = \lim_(h \to 0) (x^2 + 2hx + h^2 + x + h - 3 - x^2 - x + 3)/(h)
  3. Combine like terms:
    \displaystyle f'(x) = \lim_(h \to 0) (2hx + h^2 + h)/(h)
  4. Factor:
    \displaystyle f'(x) = \lim_(h \to 0) (h(2x + h + 1))/(h)
  5. Simplify:
    \displaystyle f'(x) = \lim_(h \to 0) 2x + h + 1
  6. Evaluate [Limit Rule - Variable Direct Substitution]:
    \displaystyle f'(x) = 2x + 0 + 1
  7. Simplify:
    \displaystyle f'(x) = 2x + 1

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

Unit: Differentiation

User Ralph Yozzo
by
9.3k points

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