Question

Differentiate implicitly to find \(d y / d x\). \(\ln \sqrt{x^{2}+y^{2}}+x y=4\)

Short answer

The derivative of the function, \(dy/dx\), is given by \(-P(x, y)/Q(x, y)\)

Step-by-step solution

Differentiate the Equation

The first step is to differentiate both sides of the equation with respect to x. Each term will thus be differentiated following the rules of calculus. Begin with the left side of the equation. Differentiate the first term, which is a natural logarithm, \(\ln(\sqrt{x^2 + y^2})\). Apply the chain rule here, where \((u(x))^n\)' = n*u^(n-1)*u(x)'. To do this, treat \(\sqrt{x^2 + y^2}\) as the inner function u. Hence, \((\ln(u))' = 1/u * u'\), Where u = \(\sqrt{x^2 + y^2}\). Differentiate the second part \(x*y\), apply the product rule for differentiation. (If the factors are u and v, then \((u*v)'\) = \(u'*v + u*v'\), where \(u'\) and \(v'\) are the derivatives of u and v, respectively), therefore \(y + x(dy/dx)\). Differentiate the right side as well, which is the constant 4. The derivative of a constant is zero hence \((4)' = 0\)

Simplify the Equation

The next step is to simplify the equation to form a single equation in form \(dy/dx = F(x, y)\). Isolate \(dy/dx\) on one side of the equation. To do this, ensure the sign integrity of the equation is maintained. The simplification results in the solution for \(dy/dx\).

Perform Algebraic Operations

For the equations \((1/2)\) * \((1/\sqrt{x^2 + y^2})\) * \((2x + 2y*dy/dx)\)+ y + x\(dy/dx\) = 0, perform algebraic operations to simplify the equation to a form \(P(x, y) + Q(x, y) dy/dx = 0\) . This is achieved by multiplying out and rearranging terms.

Solve for dy/dx

For the simplified equation, \(P(x, y) + Q(x, y)*dy/dx = 0\), solve for \(dy/dx = -P(x, y)/Q(x, y)\). This gives the required differentiation of the initial equation.