We are given that (dy)/(dx)=(sin(y))/(x). Find an expression for (d^(2)y)/(dx^(2)) in terms of x and y. (d^(2)y)/(dx^(2))=

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We are given that  (dy)/(dx)=(sin(y))/(x). Find an expression for  (d^(2)y)/(dx^(2)) in terms of  x and  y.  (d^(2)y)/(dx^(2))=

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Differentiate First Derivative: To find the second derivative of y with respect to x, we need to differentiate the first derivative with respect to x. The first derivative is given by (dy)/(dx) = (sin(y))/x. We will use the quotient rule and the chain rule to differentiate this expression.Apply Quotient Rule: The quotient rule states that for a function h(x) = f(x)/g(x), the derivative h'(x) is given by h'(x) = (g(x)f'(x) - f(x)g'(x))/(g(x))^2. In our case, f(x) = sin(y) and g(x) = x. We also need to remember that when we differentiate f(x) with respect to x, we need to use the chain rule because y is a function of x.Differentiate sin(y): First, we differentiate f(x) = sin(y) with respect to x. Using the chain rule, we get f'(x) = cos(y) * (dy)/(dx). We already know that (dy)/(dx) = (sin(y))/x, so we can substitute this in to get f'(x) = cos(y) * (sin(y))/x.Differentiate x: Next, we differentiate g(x) = x with respect to x. The derivative of x with respect to x is 1, so g'(x) = 1.Apply Quotient Rule: Now we apply the quotient rule. We have h'(x) = (x * cos(y) * (sin(y))/x - sin(y) * 1)/(x^2). This simplifies to h'(x) = (cos(y) * sin(y) - sin(y))/(x^2).Factor out sin(y): We can factor out sin(y) from the numerator to get h'(x) = sin(y) * (cos(y) - 1)/(x^2).Final Second Derivative: This expression, h'(x), is the second derivative of y with respect to x, which we denote as (d^2y)/(dx^2). Therefore, (d^2y)/(dx^2) = sin(y) * (cos(y) - 1)/(x^2).

We are given that $\frac{d y}{d x}=\frac{\sin (y)}{x}$ . $\newline$ Find an expression for $\frac{d^{2} y}{d x^{2}}$ in terms of $x$ and $y$ . $\newline$ $\frac{d^{2} y}{d x^{2}}=$

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We are given that $\frac{d y}{d x}=\frac{\sin (y)}{x}$ . $\newline$ Find an expression for $\frac{d^{2} y}{d x^{2}}$ in terms of $x$ and $y$ . $\newline$ $\frac{d^{2} y}{d x^{2}}=$