Given the function y=(2x^(2))/(6-5x^(2)), find (dy)/(dx) in simplified form. Answer: (dy)/(dx)=

Identify Function: Identify the function that needs differentiation. We are given the function y=(2x^(2))/(6-5x^(2)). We need to find its derivative with respect to x, which is denoted as (dy)/(dx).Apply Quotient Rule: Apply the quotient rule for differentiation. The quotient rule states that if we have a function in the form of u(x)/v(x), then its derivative is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2. Here, u(x) = 2x^2 and v(x) = 6 - 5x^2.Differentiate u(x): Differentiate u(x) with respect to x. The derivative of u(x) = 2x^2 with respect to x is u'(x) = 2 * 2x = 4x.Differentiate v(x): Differentiate v(x) with respect to x. The derivative of v(x) = 6 - 5x^2 with respect to x is v'(x) = 0 - 5 * 2x = -10x.Apply Quotient Rule: Apply the quotient rule using the derivatives from steps 3 and 4. (dy)/(dx) = ((6 - 5x^2) * (4x) - (2x^2) * (-10x)) / (6 - 5x^2)^2.Simplify Expression: Simplify the expression. (dy)/(dx) = (24x - 20x^3 + 20x^3) / (36 - 60x^2 + 25x^4). Notice that -20x^3 and +20x^3 cancel each other out. (dy)/(dx) = 24x / (36 - 60x^2 + 25x^4).Further Simplify: Further simplify the expression if possible. In this case, the expression cannot be simplified further.

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Algebra 2

Write a formula for an arithmetic sequence

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Q. Given the function

y=\frac{2 x^{2}}{6-5 x^{2}}

, find

\frac{d y}{d x}

in simplified form.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function that needs differentiation. $\newline$ We are given the function $y=\frac{2x^{2}}{6-5x^{2}}$ . We need to find its derivative 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 if we have a function in the form of $\frac{u(x)}{v(x)}$ , then its derivative is given by $\frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$ . Here, $u(x) = 2x^2$ and $v(x) = 6 - 5x^2$ .
Differentiate $u(x)$ : Differentiate $u(x)$ with respect to $x$ . The derivative of $u(x) = 2x^2$ with respect to $x$ is $u'(x) = 2 \times 2x = 4x$ .
Differentiate $v(x)$ : Differentiate $v(x)$ with respect to $x$ . The derivative of $v(x) = 6 - 5x^2$ with respect to $x$ is $v'(x) = 0 - 5 \times 2x = -10x$ .
Apply Quotient Rule: Apply the quotient rule using the derivatives from steps $3$ and $4$ . $\newline$ $\frac{dy}{dx} = \frac{(6 - 5x^2) \cdot (4x) - (2x^2) \cdot (-10x)}{(6 - 5x^2)^2}$ .
Simplify Expression: Simplify the expression. $\newline$ $(\frac{dy}{dx}) = \frac{24x - 20x^3 + 20x^3}{36 - 60x^2 + 25x^4}$ . $\newline$ Notice that $-20x^3$ and $+20x^3$ cancel each other out. $\newline$ $(\frac{dy}{dx}) = \frac{24x}{36 - 60x^2 + 25x^4}$ .
Further Simplify: Further simplify the expression if possible. $\newline$ In this case, the expression cannot be simplified further.