Let g(x)=(x-5)/(x^(2)+1). g^(')(x)=

Use Quotient Rule: To find the derivative of the function g(x) = (x - 5) / (x^2 + 1), we will use the quotient rule. The quotient rule states that if you have a function h(x) = f(x) / g(x), then the derivative h'(x) is given by (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. Here, f(x) = x - 5 and g(x) = x^2 + 1.Find f'(x): First, we need to find the derivative of f(x) = x - 5, which is f'(x) = 1, since the derivative of x is 1 and the derivative of a constant is 0.Find g'(x): Next, we need to find the derivative of g(x) = x^2 + 1, which is g'(x) = 2x, since the derivative of x^2 is 2x and the derivative of a constant is 0.Apply Quotient Rule: Now we apply the quotient rule. We have f'(x) = 1 and g'(x) = 2x, so we plug these into the quotient rule formula to get g'(x) = (1 * (x^2 + 1) - (x - 5) * 2x) / (x^2 + 1)^2.Simplify Numerator: Simplify the numerator of the derivative: g'(x) = (x^2 + 1 - 2x^2 + 10x) / (x^2 + 1)^2.Combine Like Terms: Combine like terms in the numerator: g'(x) = (-x^2 + 10x + 1) / (x^2 + 1)^2.Final Answer: The derivative g'(x) is now in its simplest form, so we have our final answer.

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Let $g(x)=\frac{x-5}{x^{2}+1}$ . $\newline$ $g^{\prime}(x)=$

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Q. Let

g(x)=\frac{x-5}{x^{2}+1}

\newline

g^{\prime}(x)=

Use Quotient Rule: To find the derivative of the function $g(x) = \frac{x - 5}{x^2 + 1}$ , we will use the quotient rule. The quotient rule states that if you have a function $h(x) = \frac{f(x)}{g(x)}$ , then the derivative $h'(x)$ is given by $\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ . Here, $f(x) = x - 5$ and $g(x) = x^2 + 1$ .
Find $f'(x)$ : First, we need to find the derivative of $f(x) = x - 5$ , which is $f'(x) = 1$ , since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .
Find $g'(x)$ : Next, we need to find the derivative of $g(x) = x^2 + 1$ , which is $g'(x) = 2x$ , since the derivative of $x^2$ is $2x$ and the derivative of a constant is $0$ .
Apply Quotient Rule: Now we apply the quotient rule. We have $f'(x) = 1$ and $g'(x) = 2x$ , so we plug these into the quotient rule formula to get $g'(x) = \frac{(1 \cdot (x^2 + 1) - (x - 5) \cdot 2x)}{(x^2 + 1)^2}$ .
Simplify Numerator: Simplify the numerator of the derivative: $g'(x) = \frac{x^2 + 1 - 2x^2 + 10x}{(x^2 + 1)^2}$ .
Combine Like Terms: Combine like terms in the numerator: $g'(x) = \frac{-x^2 + 10x + 1}{(x^2 + 1)^2}$ .
Final Answer: The derivative $g'(x)$ is now in its simplest form, so we have our final answer.