What is the value of (d)/(dx)((x^(2)-2x+3)/(x+1)) at x=1 ? Choose 1 answer: (A) 1 (B) -(1)/(2) (C) -2 (D) -1

Quotient Rule Application: To find the derivative of the function (x^(2)-2x+3)/(x+1), we will use the quotient rule, which states that the derivative of a function f(x)/g(x) is given by (f'(x)g(x) - f(x)g'(x))/(g(x))^2.Derivative of Numerator: First, let's find the derivative of the numerator f(x) = x^2 - 2x + 3. Using the power rule, the derivative of x^2 is 2x, the derivative of -2x is -2, and the derivative of a constant like 3 is 0. So, f'(x) = 2x - 2.Derivative of Denominator: Now, let's find the derivative of the denominator g(x) = x + 1. The derivative of x is 1, and the derivative of a constant like 1 is 0. So, g'(x) = 1.Quotient Rule Derivative: Applying the quotient rule, the derivative of the function (x^(2)-2x+3)/(x+1) is: ((2x - 2)(x + 1) - (x^2 - 2x + 3)(1))/((x + 1)^2).Numerator Simplification: Let's simplify the numerator of the derivative: (2x^2 + 2x - 2x - 2) - (x^2 - 2x + 3) = 2x^2 - x^2 + 2x - 2x - 2 + 2x - 3 = x^2 - 5.Simplified Derivative: Now we have the simplified derivative: (x^2 - 5)/((x + 1)^2).Substitute x=1: To find the value of the derivative at x=1, we substitute x with 1 in the simplified derivative: (1^2 - 5)/((1 + 1)^2) = (1 - 5)/(2^2) = (-4)/4 = -1.Derivative Value at x=1: The value of the derivative at x=1 is -1, which corresponds to answer choice (D).

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What is the value of
(d)/(dx)((x^(2)-2x+3)/(x+1)) at
x=1 ?
Choose 1 answer:
(A) 1
(B)
-(1)/(2)
(C) -2
(D) -1

What is the value of $\frac{d}{d x}\left(\frac{x^{2}-2 x+3}{x+1}\right)$ at $x=1$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $1$ $\newline$ (B) $-\frac{1}{2}$ $\newline$ (C) $-2$ $\newline$ (D) $-1$

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Find derivatives of using multiple formulae

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Q. What is the value of

\frac{d}{d x}\left(\frac{x^{2}-2 x+3}{x+1}\right)

x=1

\newline

Choose

1

answer:

\newline

(A)

1

\newline

(B)

-\frac{1}{2}

\newline

(C)

-2

\newline

(D)

-1

Quotient Rule Application: To find the derivative of the function $(x^{2}-2x+3)/(x+1)$ , we will use the quotient rule, which states that the derivative of a function $f(x)/g(x)$ is given by $(f'(x)g(x) - f(x)g'(x))/(g(x))^{2}$ .
Derivative of Numerator: First, let's find the derivative of the numerator $f(x) = x^2 - 2x + 3$ . Using the power rule, the derivative of $x^2$ is $2x$ , the derivative of $-2x$ is $-2$ , and the derivative of a constant like $3$ is $0$ . So, $f'(x) = 2x - 2$ .
Derivative of Denominator: Now, let's find the derivative of the denominator $g(x) = x + 1$ . The derivative of $x$ is $1$ , and the derivative of a constant like $1$ is $0$ . So, $g'(x) = 1$ .
Quotient Rule Derivative: Applying the quotient rule, the derivative of the function $(x^{2}-2x+3)/(x+1)$ is: $((2x - 2)(x + 1) - (x^2 - 2x + 3)(1))/((x + 1)^2)$ .
Numerator Simplification: Let's simplify the numerator of the derivative: $\newline$ egin{equation} $\newline$ ( $2$ x^ $2$ + $2$ x - $2$ x - $2$ ) - (x^ $2$ - $2$ x + $3$ ) = $2$ x^ $2$ - x^ $2$ + $2$ x - $2$ x - $2$ + $2$ x - $3$ = x^ $2$ - $5$ . $\newline$ egin{equation}
Simplified Derivative: Now we have the simplified derivative: $\frac{x^2 - 5}{(x + 1)^2}$ .
Substitute $x=1$ : To find the value of the derivative at $x=1$ , we substitute $x$ with $1$ in the simplified derivative: $\frac{1^2 - 5}{(1 + 1)^2} = \frac{1 - 5}{2^2} = \frac{-4}{4} = -1$ .
Derivative Value at $x=1$ : The value of the derivative at $x=1$ is $-1$ , which corresponds to answer choice (D).