Let y=(x)/(sin(x)). (dy)/(dx)=

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Let  y=(x)/(sin(x)).  (dy)/(dx)=

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Identify Function and Rule: Identify the function that needs differentiation and the rule that should be applied. The function y = x/sin(x) can be rewritten as y = x * sin(x)^(-1). To differentiate this function, we will use the product rule, which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Differentiate First Function: Differentiate the first function, which is x. The derivative of x with respect to x is 1.Differentiate Second Function: Differentiate the second function, which is sin(x)^(-1). The derivative of sin(x)^(-1) with respect to x is -cos(x)/sin(x)^2, which is derived using the chain rule and the derivative of sin(x), which is cos(x).Apply Product Rule: Apply the product rule. The derivative of y with respect to x is (1 * sin(x)^(-1)) + (x * (-cos(x)/sin(x)^2)).Simplify Expression: Simplify the expression. The derivative of y with respect to x is sin(x)^(-1) - xcos(x)/sin(x)^2.Combine Terms: Combine terms over a common denominator to further simplify the expression. The derivative of y with respect to x is (1 - xcos(x))/sin(x)^2.

Let $y=\frac{x}{\sin (x)}$ . $\newline$ $\frac{d y}{d x}=$

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Let $y=\frac{x}{\sin (x)}$ . $\newline$ $\frac{d y}{d x}=$