(d)/(dx)((2x+3)/((x-4)^(2)))=

Apply Quotient Rule: To find the derivative of the function (2x+3)/((x-4)^(2)), we use the quotient rule which states that if f(x) = g(x)/h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2. Here, g(x) = 2x + 3 and h(x) = (x - 4)^2.Find Derivative of g(x): First, find the derivative of g(x) = 2x + 3. Using the power rule, g'(x) = 2.Find Derivative of h(x): Next, find the derivative of h(x) = (x - 4)^2. Using the power rule and chain rule, h'(x) = 2(x - 4) * 1 = 2(x - 4).Simplify Numerator: Apply the quotient rule: f'(x) = (2 * (x - 4)^2 - (2x + 3) * 2(x - 4)) / ((x - 4)^4).Expand and Simplify: Simplify the numerator: f'(x) = (2(x - 4)^2 - (4x + 6)(x - 4)) / ((x - 4)^4).Combine Like Terms: Expand and simplify the terms in the numerator: f'(x) = (2x^2 - 16x + 32 - (4x^2 - 16x - 6x + 24)) / ((x - 4)^4).Combine like terms: f'(x) = (2x^2 - 16x + 32 - 4x^2 + 16x + 6x - 24) / ((x - 4)^4). f'(x) = (-2x^2 + 6x + 8) / ((x - 4)^4).

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$\frac{d}{dx}\left(\frac{2x+3}{(x-4)^{2}}\right)=$

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Algebra 2

Csc, sec, and cot of special angles

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\frac{d}{dx}\left(\frac{2x+3}{(x-4)^{2}}\right)=

Apply Quotient Rule: To find the derivative of the function $(2x+3)/((x-4)^{2})$ , we use the quotient rule which states that if $f(x) = g(x)/h(x)$ , then $f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^{2}$ . Here, $g(x) = 2x + 3$ and $h(x) = (x - 4)^{2}$ .
Find Derivative of $g(x)$ : First, find the derivative of $g(x) = 2x + 3$ . Using the power rule, $g'(x) = 2$ .
Find Derivative of $h(x)$ : Next, find the derivative of $h(x) = (x - 4)^2$ . Using the power rule and chain rule, $h'(x) = 2(x - 4) \cdot 1 = 2(x - 4)$ .
Simplify Numerator: Apply the quotient rule: $f'(x) = \frac{2 \cdot (x - 4)^2 - (2x + 3) \cdot 2(x - 4)}{(x - 4)^4}$ .
Expand and Simplify: Simplify the numerator: $\newline$ $f'(x) = \frac{2(x - 4)^2 - (4x + 6)(x - 4)}{(x - 4)^4}$ .
Combine Like Terms: Expand and simplify the terms in the numerator: $\newline$ $f'(x) = \frac{(2x^2 - 16x + 32 - (4x^2 - 16x - 6x + 24))}{((x - 4)^4)}.$
Combine Like Terms: Expand and simplify the terms in the numerator: $\newline$ $f'(x) = \frac{2x^2 - 16x + 32 - (4x^2 - 16x - 6x + 24)}{(x - 4)^4}$ .Combine like terms: $\newline$ $f'(x) = \frac{2x^2 - 16x + 32 - 4x^2 + 16x + 6x - 24}{(x - 4)^4}$ . $\newline$ $f'(x) = \frac{-2x^2 + 6x + 8}{(x - 4)^4}$ .