Given the function y=(6+x^(2))/(3x+2), find (dy)/(dx) in simplified form. Answer: (dy)/(dx)=

Identify Function: Identify the function to differentiate. We are given the function y=(6+x^2)/(3x+2). We need to find the derivative of this function with respect to x, which is denoted as (dy)/(dx).Apply Quotient Rule: Apply the quotient rule for differentiation. The quotient rule states that the derivative of a function in the form of u(x)/v(x) is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2. Here, u(x) = 6 + x^2 and v(x) = 3x + 2.Differentiate u(x) and v(x): Differentiate u(x) and v(x) with respect to x. The derivative of u(x) = 6 + x^2 with respect to x is u'(x) = 0 + 2x. The derivative of v(x) = 3x + 2 with respect to x is v'(x) = 3.Apply Derivatives to Quotient Rule: Apply the derivatives found in Step 3 to the quotient rule. (dy)/(dx) = ((3x + 2) * (2x) - (6 + x^2) * (3)) / (3x + 2)^2Simplify Expression: Simplify the expression. (dy)/(dx) = (6x^2 + 4x - 18 - 3x^2) / (9x^2 + 12x + 4) (dy)/(dx) = (3x^2 + 4x - 18) / (9x^2 + 12x + 4)Check for Factorization: Check for any possible simplification or factorization. The numerator and the denominator do not have any common factors, so the expression is already in its simplest form.

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Given the function
y=(6+x^(2))/(3x+2), find
(dy)/(dx) in simplified form.
Answer:
(dy)/(dx)=

Given the function $y=\frac{6+x^{2}}{3 x+2}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$

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Math Problems

Algebra 1

Write variable expressions for arithmetic sequences

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Q. Given the function

y=\frac{6+x^{2}}{3 x+2}

, find

\frac{d y}{d x}

in simplified form.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y=\frac{6+x^2}{3x+2}$ . We need to find the derivative of this function with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .
Apply Quotient Rule: Apply the quotient rule for differentiation. $\newline$ The quotient rule states that the derivative of a function in the form of $\frac{u(x)}{v(x)}$ is given by $\frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$ . Here, $u(x) = 6 + x^2$ and $v(x) = 3x + 2$ .
Differentiate $u(x)$ and $v(x)$ : Differentiate $u(x)$ and $v(x)$ with respect to $x$ . The derivative of $u(x) = 6 + x^2$ with respect to $x$ is $u'(x) = 0 + 2x$ . The derivative of $v(x) = 3x + 2$ with respect to $x$ is $v(x)$ $0$ .
Apply Derivatives to Quotient Rule: Apply the derivatives found in Step $3$ to the quotient rule. $\frac{dy}{dx} = \frac{(3x + 2) \cdot (2x) - (6 + x^2) \cdot (3)}{(3x + 2)^2}$
Simplify Expression: Simplify the expression. $\newline$ $(\frac{dy}{dx}) = \frac{(6x^2 + 4x - 18 - 3x^2)}{(9x^2 + 12x + 4)}$ $\newline$ $(\frac{dy}{dx}) = \frac{(3x^2 + 4x - 18)}{(9x^2 + 12x + 4)}$
Check for Factorization: Check for any possible simplification or factorization. The numerator and the denominator do not have any common factors, so the expression is already in its simplest form.