Given the function y=(1+x^(4))/(5x-1), find (dy)/(dx) in simplified form. Answer: (dy)/(dx)=

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Given the function  y=(1+x^(4))/(5x-1), find  (dy)/(dx) in simplified form. Answer:  (dy)/(dx)=

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Apply Quotient Rule: To find the derivative of the function y=(1+x^(4))/(5x-1) with respect to x, we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions, u(x)/v(x), then its derivative is given by (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2. Here, u(x) = 1 + x^4 and v(x) = 5x - 1.Find u(x): First, we need to find the derivative of u(x) = 1 + x^4 with respect to x. The derivative of a constant is 0, and the derivative of x^4 with respect to x is 4x^3. Therefore, u'(x) = 0 + 4x^3 = 4x^3.Find v(x): Next, we need to find the derivative of v(x) = 5x - 1 with respect to x. The derivative of 5x with respect to x is 5, and the derivative of a constant is 0. Therefore, v'(x) = 5 - 0 = 5.Plug into Quotient Rule: Now we apply the quotient rule. We have u(x) = 1 + x^4, v(x) = 5x - 1, u'(x) = 4x^3, and v'(x) = 5. Plugging these into the quotient rule formula, we get:  (dy)/(dx) = ((5x - 1) * (4x^3) - (1 + x^4) * (5)) / (5x - 1)^2.Simplify Numerator: We simplify the numerator of the derivative:  (5x - 1) * (4x^3) - (1 + x^4) * (5) = 20x^4 - 4x^3 - 5 - 5x^4.Combine Like Terms: Combine like terms in the numerator:  20x^4 - 4x^3 - 5 - 5x^4 = 15x^4 - 4x^3 - 5.Final Derivative: Now we have the simplified form of the derivative:  (dy)/(dx) = (15x^4 - 4x^3 - 5) / (5x - 1)^2.

Given the function $y=\frac{1+x^{4}}{5 x-1}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$

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Given the function $y=\frac{1+x^{4}}{5 x-1}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$