we are given that $\\frac{dy}{dx}=\\frac{\\sin(y)}{x}$. find an expression for $\\frac{d^{2}y}{dx^{2}}$ in…

we are given that $\\frac{dy}{dx}=\\frac{\\sin(y)}{x}$. find an expression for $\\frac{d^{2}y}{dx^{2}}$ in terms of $x$ and $y$. $\\frac{d^{2}y}{dx^{2}}=\\square$

we are given that $\\frac{dy}{dx}=\\frac{\\sin(y)}{x}$. find an expression for $\\frac{d^{2}y}{dx^{2}}$ in terms of $x$ and $y$. $\\frac{d^{2}y}{dx^{2}}=\\square$

Answer

Explanation:

Step1: Differentiate both sides with respect to (x)

Use the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}), where (u = \sin(y)) and (v=x). By the chain - rule, (\frac{d}{dx}(\sin(y))=\cos(y)\frac{dy}{dx}). So, (\frac{d^{2}y}{dx^{2}}=\frac{\cos(y)(\frac{dy}{dx})\cdot x-\sin(y)\cdot1}{x^{2}})

Step2: Substitute (\frac{dy}{dx}=\frac{\sin(y)}{x}) into the above formula

Substitute (\frac{dy}{dx}) into (\frac{d^{2}y}{dx^{2}}=\frac{\cos(y)(\frac{dy}{dx})\cdot x-\sin(y)\cdot1}{x^{2}}) We get (\frac{d^{2}y}{dx^{2}}=\frac{\cos(y)\cdot\frac{\sin(y)}{x}\cdot x-\sin(y)}{x^{2}})

Step3: Simplify the expression

Simplify (\frac{\cos(y)\cdot\frac{\sin(y)}{x}\cdot x-\sin(y)}{x^{2}}) (\frac{d^{2}y}{dx^{2}}=\frac{\sin(y)\cos(y)-\sin(y)}{x^{2}}=\frac{\sin(y)(\cos(y) - 1)}{x^{2}})

Answer:

(\frac{\sin(y)(\cos(y)-1)}{x^{2}})