If y^(2)cos((1)/(x))=a^(2), then find (dy)/(dx)

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If  y^(2)cos((1)/(x))=a^(2), then find  (dy)/(dx)

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Differentiate with respect to x: Differentiate both sides of the equation with respect to x. We have the equation y^(2)cos((1)/(x))=a^(2). To find (dy)/(dx), we need to differentiate both sides of the equation with respect to x. Since a^(2) is a constant, its derivative is 0.Apply product rule: Apply the product rule to the left side of the equation. The left side of the equation is a product of y^(2) and cos((1)/(x)). The product rule 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 y^2: Differentiate y^(2) with respect to x. The derivative of y^(2) with respect to x is 2y(dy)/(dx), using the chain rule.Differentiate cos(1/x): Differentiate cos((1)/(x)) with respect to x. The derivative of cos((1)/(x)) with respect to x is sin((1)/(x)) multiplied by the derivative of (1)/(x), which is -1/x^2, again using the chain rule.Write down differentiated equation: Write down the differentiated equation. After applying the product rule and differentiating each part, we get: 2y(dy)/(dx) * cos((1)/(x)) - y^(2) * sin((1)/(x)) * (-1/x^2) = 0Solve for dy/dx: Solve for (dy)/(dx). To solve for (dy)/(dx), we need to isolate it on one side of the equation. We can do this by adding y^(2) * sin((1)/(x)) * (1/x^2) to both sides and then dividing by 2y * cos((1)/(x)).Simplify to find dy/dx: Simplify the equation to find (dy)/(dx). (dy)/(dx) = (y^(2) * sin((1)/(x)) * (1/x^2)) / (2y * cos((1)/(x))) (dy)/(dx) = (y * sin((1)/(x)) * (1/x^2)) / (2 * cos((1)/(x)))

If $y^{2}\cos\left(\frac{1}{x}\right)=a^{2}$ , then find $\frac{dy}{dx}$

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If $y^{2}\cos\left(\frac{1}{x}\right)=a^{2}$ , then find $\frac{dy}{dx}$