Given the function y=(6+x^(4))/(5x-2), find (dy)/(dx) in simplified form.

Identify function: Identify the function to differentiate. We are given the function y=(6+x^(4))/(5x-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, where u(x) is the numerator and v(x) is the denominator of the function.Differentiate numerator and denominator: Differentiate the numerator and the denominator separately. The numerator u(x) = 6 + x^4, and its derivative u'(x) is 0 + 4x^3. The denominator v(x) = 5x - 2, and its derivative v'(x) is 5.Apply quotient rule with derivatives: Apply the quotient rule using the derivatives from Step 3. (dy)/(dx) = ((5x - 2) * (4x^3) - (6 + x^4) * 5) / (5x - 2)^2Expand numerator: Expand the numerator of the derivative. (dy)/(dx) = (20x^4 - 8x^3 - 5x^4 - 30) / (5x - 2)^2Combine like terms: Combine like terms in the numerator. (dy)/(dx) = (15x^4 - 8x^3 - 30) / (5x - 2)^2Simplify derivative expression: Simplify the derivative expression if possible. The expression is already simplified, so this is the final form of the derivative. (dy)/(dx) = (15x^4 - 8x^3 - 30) / (5x - 2)^2

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Given the function $\newline$ $y=\frac{6+x^{4}}{5x-2}$ , find $\newline$ $\frac{dy}{dx}$ in simplified form.

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Q. Given the function

\newline

y=\frac{6+x^{4}}{5x-2}

, find

\newline

\frac{dy}{dx}

in simplified form.

Identify function: Identify the function to differentiate. $\newline$ We are given the function $y=\frac{6+x^{4}}{5x-2}$ . 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 $u(x)/v(x)$ is given by $(v(x) \cdot u'(x) - u(x) \cdot v'(x)) / (v(x))^2$ , where $u(x)$ is the numerator and $v(x)$ is the denominator of the function.
Differentiate numerator and denominator: Differentiate the numerator and the denominator separately. $\newline$ The numerator $u(x) = 6 + x^4$ , and its derivative $u'(x)$ is $0 + 4x^3$ . $\newline$ The denominator $v(x) = 5x - 2$ , and its derivative $v'(x)$ is $5$ .
Apply quotient rule with derivatives: Apply the quotient rule using the derivatives from Step $3$ . $\newline$ $\frac{dy}{dx} = \frac{(5x - 2) \cdot (4x^3) - (6 + x^4) \cdot 5}{(5x - 2)^2}$
Expand numerator: Expand the numerator of the derivative. $\frac{dy}{dx} = \frac{20x^4 - 8x^3 - 5x^4 - 30}{(5x - 2)^2}$
Combine like terms: Combine like terms in the numerator. $\newline$ $\frac{dy}{dx} = \frac{15x^4 - 8x^3 - 30}{(5x - 2)^2}$
Simplify derivative expression: Simplify the derivative expression if possible. $\newline$ The expression is already simplified, so this is the final form of the derivative. $\newline$ $\frac{dy}{dx} = \frac{15x^4 - 8x^3 - 30}{(5x - 2)^2}$