Engineering, 04.02.2021 04:50 QueenNerdy889

. Find the derivative of y = x² – 5x using the definition of the derivative.

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Answer from: sim2004

$\displaystyle y' = 2x - 5$

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: $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Explanation:

Step 1: Define

Identify

$\displaystyle y = x^2 - 5x$

Step 2: Differentiate

Substitute in function [Definition of a Derivative]: $\displaystyle y' = \lim_{h \to 0} \frac{[(x + h)^2 - 5(x + h)] - (x^2 - 5x)}{h}$ Expand: $\displaystyle y' = \lim_{h \to 0} \frac{x^2 + 2hx + h^2 - 5x - 5h - x^2 + 5x}{h}$ Combine like terms: $\displaystyle y' = \lim_{h \to 0} \frac{2hx + h^2 - 5h}{h}$ Factor: $\displaystyle y' = \lim_{h \to 0} \frac{h(2x + h - 5)}{h}$ Simplify: $\displaystyle y' = \lim_{h \to 0} (2x + h - 5)$ Evaluate limit [Limit Rule - Variable Direct Substitution]: $\displaystyle y' = 2x + 0 - 5$ Simplify: $\displaystyle y' = 2x - 5$

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

Unit: Differentiation

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. Find the derivative of y = x² – 5x using the definition of the derivative....

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