(4x^(2)+1)/(x^(3)-4x)

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Quotient Rule: To find the derivative of the function (4x^(2)+1)/(x^(3)-4x), we will use the quotient rule. The quotient rule states that if we have a function h(x) = f(x)/g(x), then its derivative h'(x) is given by (f'(x)g(x) - f(x)g'(x))/(g(x))^2.Identify Functions: First, let's identify the numerator f(x) = 4x^2 + 1 and the denominator g(x) = x^3 - 4x. We will need to find the derivatives f'(x) and g'(x) of these two functions.Derivative of Numerator: The derivative of the numerator f(x) = 4x^2 + 1 is f'(x) = 8x, since the derivative of x^2 is 2x and the derivative of a constant is 0.Derivative of Denominator: The derivative of the denominator g(x) = x^3 - 4x is g'(x) = 3x^2 - 4, since the derivative of x^3 is 3x^2 and the derivative of 4x is 4.Apply Quotient Rule: Now we apply the quotient rule: h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2. Plugging in the derivatives we found, we get h'(x) = (8x(x^3 - 4x) - (4x^2 + 1)(3x^2 - 4))/(x^3 - 4x)^2.Simplify Numerator: We simplify the numerator of the derivative: 8x(x^3 - 4x) - (4x^2 + 1)(3x^2 - 4) = 8x^4 - 32x^2 - (12x^4 - 16x^2 + 3x^2 - 4).Combine Like Terms: Further simplifying the numerator, we combine like terms: 8x^4 - 32x^2 - 12x^4 + 16x^2 - 3x^2 + 4 = -4x^4 - 19x^2 + 4.Final Derivative: Now we have the simplified form of the derivative: h'(x) = (-4x^4 - 19x^2 + 4)/(x^3 - 4x)^2.

$\frac{4 x^{2}+1}{x^{3}-4 x}$

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4x2+1x3−4x \frac{4 x^{2}+1}{x^{3}-4 x} x3−4x4x2+1​

$\frac{4 x^{2}+1}{x^{3}-4 x}$