Let y=(cos(x))/(x^(3)). (dy)/(dx)=

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Let  y=(cos(x))/(x^(3)).  (dy)/(dx)=

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Identify Functions: We are given the function y = (cos(x))/(x^(3)). To find the derivative of y with respect to x, we will use the quotient rule, which states that the derivative of a function in the form of f(x)/g(x) is (f'(x)g(x) - f(x)g'(x))/(g(x))^2.Find Derivatives: First, identify the functions f(x) and g(x) where f(x) = cos(x) and g(x) = x^(3). Then find their derivatives f'(x) and g'(x). The derivative of f(x) = cos(x) with respect to x is f'(x) = -sin(x). The derivative of g(x) = x^(3) with respect to x is g'(x) = 3x^(2).Apply Quotient Rule: Now apply the quotient rule. The derivative of y with respect to x is: (dy)/(dx) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2 Substitute f(x), f'(x), g(x), and g'(x) into the formula: (dy)/(dx) = ((-sin(x))(x^(3)) - (cos(x))(3x^(2)))/(x^(3))^2Simplify Expression: Simplify the expression by distributing and combining like terms: (dy)/(dx) = (-x^(3)sin(x) - 3x^(2)cos(x))/(x^(6))Factor Out x^2: We can further simplify the expression by factoring out an x^2 from the numerator: (dy)/(dx) = (x^2(-xsin(x) - 3cos(x)))/(x^(6))Cancel Out x^2: Now, we can cancel out an x^2 from the numerator and denominator, as long as x is not equal to 0 (since we cannot divide by zero): (dy)/(dx) = (-xsin(x) - 3cos(x))/(x^(4))

Let $y=\frac{\cos (x)}{x^{3}}$ . $\newline$ $\frac{d y}{d x}=$

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Let $y=\frac{\cos (x)}{x^{3}}$ . $\newline$ $\frac{d y}{d x}=$