Given the function y=(3-x^(4))/(1-2x^(4)), find (dy)/(dx) in simplified form.

Question

Given the function  y=(3-x^(4))/(1-2x^(4)), find  (dy)/(dx) in simplified form.

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Identify u and v: Given the function y=(3-x^(4))/(1-2x^(4)), we need to find the derivative of y with respect to x, which is denoted as (dy)/(dx). We will use the quotient rule for derivatives, which states that if y = u/v, then (dy)/(dx) = (v(du/dx) - u(dv/dx))/(v^2), where u and v are functions of x.Find (du/dx): First, let's identify u and v in our function: u = 3 - x^4 and v = 1 - 2x^4. Now we need to find the derivatives of u and v with respect to x, which are denoted as (du/dx) and (dv/dx) respectively.Find (dv/dx): To find (du/dx), we differentiate u = 3 - x^4 with respect to x. (du/dx) = 0 - 4x^3 = -4x^3.Apply quotient rule: To find (dv/dx), we differentiate v = 1 - 2x^4 with respect to x. (dv/dx) = 0 - 8x^3 = -8x^3.Simplify the numerator: Now we apply the quotient rule: (dy)/(dx) = (v(du/dx) - u(dv/dx))/(v^2). Substituting the values we have: (dy)/(dx) = ((1 - 2x^4)(-4x^3) - (3 - x^4)(-8x^3))/((1 - 2x^4)^2).Further simplify if possible: We simplify the numerator: (dy)/(dx) = (-4x^3 + 8x^7 - 24x^3 + 8x^7)/((1 - 2x^4)^2). (dy)/(dx) = (16x^7 - 28x^3)/((1 - 2x^4)^2).Final derivative: We simplify the expression further if possible. However, in this case, the expression is already in its simplest form. So, the derivative of the function y with respect to x is: (dy)/(dx) = (16x^7 - 28x^3)/((1 - 2x^4)^2).

Given the function $\newline$ $y=\frac{3-x^{4}}{1-2x^{4}},$ find $\newline$ $\frac{dy}{dx}$ in simplified form.

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Given the function $\newline$ $y=\frac{3-x^{4}}{1-2x^{4}},$ find $\newline$ $\frac{dy}{dx}$ in simplified form.