$\frac{d}{d x}\left[\sqrt{8 x^{2}+2 x-3}\right]=?$ $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{8 x+1}{\sqrt{8 x^{2}+2 x-3}}$ $\newline$ (B) $\frac{8 x+1}{\sqrt{x}}$ $\newline$ (c) $\sqrt{16 x+2}$ $\newline$ (D) $\frac{1}{2 \sqrt{16 x+2}}$

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Algebra 2

Simplify rational expressions

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\frac{d}{d x}\left[\sqrt{8 x^{2}+2 x-3}\right]=?

\newline

Choose

1

answer:

\newline

(A)

\frac{8 x+1}{\sqrt{8 x^{2}+2 x-3}}

\newline

(B)

\frac{8 x+1}{\sqrt{x}}

\newline

(c)

\sqrt{16 x+2}

\newline

(D)

\frac{1}{2 \sqrt{16 x+2}}

Identify function: Identify the function to differentiate. $\newline$ We are given the function $f(x) = \sqrt{8x^2 + 2x - 3}$ . We need to find its derivative with respect to $x$ , which is denoted as $\frac{d}{dx}[f(x)]$ .
Apply chain rule: Apply the chain rule for differentiation. $\newline$ The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is the square root function and the inner function is the quadratic polynomial $8x^2 + 2x - 3$ .
Differentiate outer function: Differentiate the outer function. $\newline$ The derivative of $\sqrt{u}$ with respect to $u$ is $\frac{1}{2\sqrt{u}}$ . So, the derivative of the outer function with respect to the inner function is $\frac{1}{2\sqrt{8x^2 + 2x - 3}}$ .
Differentiate inner function: Differentiate the inner function. $\newline$ The derivative of $8x^2 + 2x - 3$ with respect to $x$ is $16x + 2$ .
Apply chain rule multiplication: Apply the chain rule by multiplying the derivatives of the outer and inner functions. $\newline$ The derivative of the function $f(x)$ with respect to $x$ is $\frac{1}{2\sqrt{8x^2 + 2x - 3}}$ $\cdot$ ( $16x + 2$ ).
Simplify expression: Simplify the expression. $\newline$ We can simplify the expression by multiplying the numerator of the first fraction by the second fraction. This gives us $(16x + 2)/(2\sqrt{8x^2 + 2x - 3})$ .
Further simplify expression: Further simplify the expression by dividing the numerator by $2$ . This gives us $\frac{8x + 1}{\sqrt{8x^2 + 2x - 3}}$ .
Match with answer choices: Match the simplified expression with the given answer choices. $\newline$ The simplified expression $(8x + 1)/\sqrt{8x^2 + 2x - 3}$ matches with answer choice (A).