(d)/(dx)((3x^(2)-1)/(x-2))=

Use Quotient Rule: To find the derivative of the function (3x^2 - 1)/(x - 2), we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions, f(x)/g(x), its derivative is given by (f'(x)g(x) - f(x)g'(x))/(g(x))^2.Identify Functions: First, let's identify the functions f(x) and g(x) in our case. Here, f(x) = 3x^2 - 1 and g(x) = x - 2. We will need to find their derivatives, f'(x) and g'(x), respectively.Find Derivatives: The derivative of f(x) = 3x^2 - 1 with respect to x is f'(x) = 6x, since the derivative of x^2 is 2x and the derivative of a constant is 0.Apply Quotient Rule: The derivative of g(x) = x - 2 with respect to x is g'(x) = 1, since the derivative of x is 1 and the derivative of a constant is 0.Simplify Numerator: Now we apply the quotient rule. The derivative of our function is: (6x * (x - 2) - (3x^2 - 1) * 1) / (x - 2)^2.Combine Like Terms: Let's simplify the numerator of the derivative. We distribute 6x into (x - 2) and (3x^2 - 1) into 1: (6x^2 - 12x - (3x^2 - 1)) / (x - 2)^2.Final Simplified Form: Further simplifying the numerator, we combine like terms: (6x^2 - 12x - 3x^2 + 1) / (x - 2)^2.After combining like terms, the numerator becomes: (3x^2 - 12x + 1) / (x - 2)^2.This is the simplified form of the derivative. There are no further simplifications, and we have not made any math errors.

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

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Algebra 2

Csc, sec, and cot of special angles

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

Use Quotient Rule: To find the derivative of the function $(3x^2 - 1)/(x - 2)$ , we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions, $f(x)/g(x)$ , its derivative is given by $(f'(x)g(x) - f(x)g'(x))/(g(x))^2$ .
Identify Functions: First, let's identify the functions $f(x)$ and $g(x)$ in our case. Here, $f(x) = 3x^2 - 1$ and $g(x) = x - 2$ . We will need to find their derivatives, $f'(x)$ and $g'(x)$ , respectively.
Find Derivatives: The derivative of $f(x) = 3x^2 - 1$ with respect to $x$ is $f'(x) = 6x$ , since the derivative of $x^2$ is $2x$ and the derivative of a constant is $0$ .
Apply Quotient Rule: The derivative of $g(x) = x - 2$ with respect to $x$ is $g'(x) = 1$ , since the derivative of $x$ is $1$ and the derivative of a constant is $0$ .
Simplify Numerator: Now we apply the quotient rule. The derivative of our function is: $\newline$ egin{equation} $\newline$ \frac{( $6$ x \cdot (x - $2$ ) - ( $3$ x^ $2$ - $1$ ) \cdot $1$ )}{(x - $2$ )^ $2$ }. $\newline$ \end{equation}
Combine Like Terms: Let's simplify the numerator of the derivative. We distribute $6x$ into $(x - 2)$ and $(3x^2 - 1)$ into $1$ : $(6x^2 - 12x - (3x^2 - 1)) / (x - 2)^2.$
Final Simplified Form: Further simplifying the numerator, we combine like terms: $\newline$ egin{equation} $\newline$ \frac{( $6$ x^ $2$ - $12$ x - $3$ x^ $2$ + $1$ )}{(x - $2$ )^ $2$ }. $\newline$ \end{equation}
Final Simplified Form: Further simplifying the numerator, we combine like terms: $\newline$ $(6x^2 - 12x - 3x^2 + 1) / (x - 2)^2$ .After combining like terms, the numerator becomes: $\newline$ $(3x^2 - 12x + 1) / (x - 2)^2$ .
Final Simplified Form: Further simplifying the numerator, we combine like terms: $\newline$ $(6x^2 - 12x - 3x^2 + 1) / (x - 2)^2$ .After combining like terms, the numerator becomes: $\newline$ $(3x^2 - 12x + 1) / (x - 2)^2$ .This is the simplified form of the derivative. There are no further simplifications, and we have not made any math errors.