(d)/(dx)((2x^(2)+x-3)/(2x+7))=

Identify Function: Identify the function to differentiate. The function given is a quotient of two functions, which means we will use the quotient rule to find the derivative. The quotient rule states that the derivative of a function h(x) = f(x)/g(x) is given by h'(x) = (g(x)f'(x) - f(x)g'(x)) / (g(x))^2.Differentiate Numerator and Denominator: Differentiate the numerator and the denominator separately. The numerator is 2x^2 + x - 3, and its derivative is 4x + 1. The denominator is 2x + 7, and its derivative is 2.Apply Quotient Rule: Apply the quotient rule. Using the derivatives from Step 2, we have: h'(x) = ((2x + 7)(4x + 1) - (2x^2 + x - 3)(2)) / (2x + 7)^2.Expand Numerator: Expand the numerator of the derivative. We need to distribute the terms: (2x + 7)(4x + 1) = 8x^2 + 2x + 28x + 7 = 8x^2 + 30x + 7, (2x^2 + x - 3)(2) = 4x^2 + 2x - 6. Now subtract the second product from the first: (8x^2 + 30x + 7) - (4x^2 + 2x - 6) = 8x^2 + 30x + 7 - 4x^2 - 2x + 6 = 4x^2 + 28x + 13.Write Simplified Derivative: Write the derivative in its simplified form. The derivative is: h'(x) = (4x^2 + 28x + 13) / (2x + 7)^2.Check Simplified Form: Check if the derivative is in the simplest form and if it answers the question prompt. The derivative is simplified, and it is the answer to the question prompt.

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$\frac{d}{d x}\left(\frac{2 x^{2}+x-3}{2 x+7}\right)=$

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Algebra 2

Csc, sec, and cot of special angles

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\frac{d}{d x}\left(\frac{2 x^{2}+x-3}{2 x+7}\right)=

Identify Function: Identify the function to differentiate. The function given is a quotient of two functions, which means we will use the quotient rule to find the derivative. The quotient rule states that the derivative of a function $h(x) = \frac{f(x)}{g(x)}$ is given by $h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$ .
Differentiate Numerator and Denominator: Differentiate the numerator and the denominator separately. The numerator is $2x^2 + x - 3$ , and its derivative is $4x + 1$ . The denominator is $2x + 7$ , and its derivative is $2$ .
Apply Quotient Rule: Apply the quotient rule. Using the derivatives from Step $2$ , we have: $\newline$ $h'(x) = \frac{(2x + 7)(4x + 1) - (2x^2 + x - 3)(2)}{(2x + 7)^2}$ .
Expand Numerator: Expand the numerator of the derivative. We need to distribute the terms: $\newline$ $(2x + 7)(4x + 1) = 8x^2 + 2x + 28x + 7 = 8x^2 + 30x + 7$ , $\newline$ $(2x^2 + x - 3)(2) = 4x^2 + 2x - 6$ . $\newline$ Now subtract the second product from the first: $\newline$ $(8x^2 + 30x + 7) - (4x^2 + 2x - 6) = 8x^2 + 30x + 7 - 4x^2 - 2x + 6 = 4x^2 + 28x + 13$ .
Write Simplified Derivative: Write the derivative in its simplified form. The derivative is: $h'(x) = \frac{4x^2 + 28x + 13}{(2x + 7)^2}$ .
Check Simplified Form: Check if the derivative is in the simplest form and if it answers the question prompt. The derivative is simplified, and it is the answer to the question prompt.