(d)/(dx)((x^(2)-x+5)/(4x-1))=

Apply Quotient Rule: To find the derivative of the function (x^2 - x + 5) / (4x - 1), we will use the quotient rule. The quotient rule states that if we have a function h(x) = f(x) / g(x), then its derivative h'(x) is given by h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. Here, f(x) = x^2 - x + 5 and g(x) = 4x - 1.Find f'(x): First, we find the derivative of f(x) = x^2 - x + 5. The derivative of x^2 is 2x, the derivative of -x is -1, and the derivative of a constant like 5 is 0. So, f'(x) = 2x - 1.Find g'(x): Next, we find the derivative of g(x) = 4x - 1. The derivative of 4x is 4, and the derivative of a constant like -1 is 0. So, g'(x) = 4.Expand Numerator: Now we apply the quotient rule. We have f'(x) = 2x - 1 and g'(x) = 4, so we plug these into the quotient rule formula: h'(x) = ((2x - 1)(4x - 1) - (x^2 - x + 5)(4)) / (4x - 1)^2.Subtract Terms: We expand the numerator: (2x - 1)(4x - 1) = 8x^2 - 2x - 4x + 1 = 8x^2 - 6x + 1, (x^2 - x + 5)(4) = 4x^2 - 4x + 20. So, the numerator becomes (8x^2 - 6x + 1) - (4x^2 - 4x + 20).Simplify Result: Subtract the terms in the numerator: (8x^2 - 6x + 1) - (4x^2 - 4x + 20) = 8x^2 - 6x + 1 - 4x^2 + 4x - 20 = 4x^2 - 2x - 19.Now we have the simplified numerator and the denominator squared: h'(x) = (4x^2 - 2x - 19) / (4x - 1)^2. This is the derivative of the function (x^2 - x + 5) / (4x - 1) in simplified form.

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Csc, sec, and cot of special angles

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\frac{d}{d x}\left(\frac{x^{2}-x+5}{4 x-1}\right)=

Apply Quotient Rule: To find the derivative of the function $(x^2 - x + 5) / (4x - 1)$ , we will use the quotient rule. The quotient rule states that if we have a function $h(x) = f(x) / g(x)$ , then its derivative $h'(x)$ is given by $h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2$ . Here, $f(x) = x^2 - x + 5$ and $g(x) = 4x - 1$ .
Find $f'(x)$ : First, we find the derivative of $f(x) = x^2 - x + 5$ . The derivative of $x^2$ is $2x$ , the derivative of $-x$ is $-1$ , and the derivative of a constant like $5$ is $0$ . So, $f'(x) = 2x - 1$ .
Find $g'(x)$ : Next, we find the derivative of $g(x) = 4x - 1$ . The derivative of $4x$ is $4$ , and the derivative of a constant like $-1$ is $0$ . So, $g'(x) = 4$ .
Expand Numerator: Now we apply the quotient rule. We have $f'(x) = 2x - 1$ and $g'(x) = 4$ , so we plug these into the quotient rule formula: $\newline$ $h'(x) = \frac{(2x - 1)(4x - 1) - (x^2 - x + 5)(4)}{(4x - 1)^2}$ .
Subtract Terms: We expand the numerator: $\newline$ $(2x - 1)(4x - 1) = 8x^2 - 2x - 4x + 1 = 8x^2 - 6x + 1$ , $\newline$ $(x^2 - x + 5)(4) = 4x^2 - 4x + 20$ . $\newline$ So, the numerator becomes $(8x^2 - 6x + 1) - (4x^2 - 4x + 20)$ .
Simplify Result: Subtract the terms in the numerator: $\newline$ egin{equation} $\newline$ ( $8$ x^ $2$ - $6$ x + $1$ ) - ( $4$ x^ $2$ - $4$ x + $20$ ) = $8$ x^ $2$ - $6$ x + $1$ - $4$ x^ $2$ + $4$ x - $20$ = $4$ x^ $2$ - $2$ x - $19$ . $\newline$ egin{equation}
Simplify Result: Subtract the terms in the numerator: $\newline$ $(8x^2 - 6x + 1) - (4x^2 - 4x + 20) = 8x^2 - 6x + 1 - 4x^2 + 4x - 20 = 4x^2 - 2x - 19$ .Now we have the simplified numerator and the denominator squared: $\newline$ $h'(x) = \frac{4x^2 - 2x - 19}{(4x - 1)^2}$ . $\newline$ This is the derivative of the function $\frac{x^2 - x + 5}{4x - 1}$ in simplified form.