Let y=(x^(2)+2x)/(4-5x). What is the value of (dy)/(dx) at x=2? Choose 1 answer: (A) -(4)/(3) (B) (1)/(9) (c) -(6)/(5) (D) -(2)/(3)

Differentiate Function: Differentiate the function y with respect to x. We have y = (x^2 + 2x) / (4 - 5x). To find dy/dx, we need to use the quotient rule for differentiation, which is (v(u') - u(v')) / v^2, where u is the numerator and v is the denominator of the function.Identify u and v: Identify u and v and their derivatives. Let u = x^2 + 2x and v = 4 - 5x. Then, the derivatives are u' = d(u)/dx = 2x + 2 and v' = d(v)/dx = -5.Apply Quotient Rule: Apply the quotient rule. Using the quotient rule, dy/dx = [(4 - 5x)(2x + 2) - (x^2 + 2x)(-5)] / (4 - 5x)^2.Simplify Expression: Simplify the expression. dy/dx = [(8x - 10x^2 + 8 - 10x) - (-5x^2 - 10x)] / (16 - 40x + 25x^2). dy/dx = (-10x^2 + 8x + 8 + 5x^2 + 10x) / (16 - 40x + 25x^2). dy/dx = (-5x^2 + 18x + 8) / (16 - 40x + 25x^2).Plug in x = 2: Plug in x = 2 into the simplified expression of dy/dx. dy/dx at x = 2 is (-5(2)^2 + 18(2) + 8) / (16 - 40(2) + 25(2)^2). dy/dx at x = 2 is (-5(4) + 36 + 8) / (16 - 80 + 100). dy/dx at x = 2 is (-20 + 36 + 8) / (16 + 20). dy/dx at x = 2 is (24) / (36). dy/dx at x = 2 is 2/3.

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Let
y=(x^(2)+2x)/(4-5x).
What is the value of
(dy)/(dx) at
x=2?
Choose 1 answer:
(A)
-(4)/(3)
(B)
(1)/(9)
(c)
-(6)/(5)
(D)
-(2)/(3)

Let $y=\frac{x^{2}+2 x}{4-5 x}$ . $\newline$ What is the value of $\frac{d y}{d x}$ at $x=2 ?$ $\newline$ Choose $1$ answer: $\newline$ (A) $-\frac{4}{3}$ $\newline$ (B) $\frac{1}{9}$ $\newline$ (c) $-\frac{6}{5}$ $\newline$ (D) $-\frac{2}{3}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Let

y=\frac{x^{2}+2 x}{4-5 x}

\newline

What is the value of

\frac{d y}{d x}

x=2 ?

\newline

Choose

1

answer:

\newline

(A)

-\frac{4}{3}

\newline

(B)

\frac{1}{9}

\newline

(c)

-\frac{6}{5}

\newline

(D)

-\frac{2}{3}

Differentiate Function: Differentiate the function $y$ with respect to $x$ . We have $y = \frac{x^2 + 2x}{4 - 5x}$ . To find $\frac{dy}{dx}$ , we need to use the quotient rule for differentiation, which is $\frac{v(u') - u(v')}{v^2}$ , where $u$ is the numerator and $v$ is the denominator of the function.
Identify $u$ and $v$ : Identify $u$ and $v$ and their derivatives. $\newline$ Let $u = x^2 + 2x$ and $v = 4 - 5x$ . Then, the derivatives are $u' = \frac{d(u)}{dx} = 2x + 2$ and $v' = \frac{d(v)}{dx} = -5$ .
Apply Quotient Rule: Apply the quotient rule. $\newline$ Using the quotient rule, $\frac{dy}{dx} = \frac{(4 - 5x)(2x + 2) - (x^2 + 2x)(-5)}{(4 - 5x)^2}$ .
Simplify Expression: Simplify the expression. $\newline$ $\frac{dy}{dx} = \frac{[8x - 10x^2 + 8 - 10x] - [-5x^2 - 10x]}{16 - 40x + 25x^2}$ . $\newline$ $\frac{dy}{dx} = \frac{-10x^2 + 8x + 8 + 5x^2 + 10x}{16 - 40x + 25x^2}$ . $\newline$ $\frac{dy}{dx} = \frac{-5x^2 + 18x + 8}{16 - 40x + 25x^2}$ .
Plug in $x = 2$ : Plug in $x = 2$ into the simplified expression of $\frac{dy}{dx}$ .
$\frac{dy}{dx}$ at $x = 2$ is $\frac{-5(2)^2 + 18(2) + 8}{16 - 40(2) + 25(2)^2}$ .
$\frac{dy}{dx}$ at $x = 2$ is $\frac{-5(4) + 36 + 8}{16 - 80 + 100}$ .
$\frac{dy}{dx}$ at $x = 2$ is $x = 2$ $1$ .
$\frac{dy}{dx}$ at $x = 2$ is $x = 2$ $4$ .
$\frac{dy}{dx}$ at $x = 2$ is $x = 2$ $7$ .