Question

Find \(d y / d x\) by implicit differentiation and evaluate the derivative at the indicated point. $$ y^{2}=\ln x, \quad(e, 1) $$

Short answer

So, the derivative of \(y\) with respect to \(x\) evaluated at \((e, 1)\) is \(\frac{1}{2e}\).

Step-by-step solution

Apply Implicit Differentiation

In the given function, both \(y\) and \(x\) are variables. So, apply implicit differentiation to both sides of the function \(y^{2} = \ln x\).The derivative of \(y^{2}\) with respect to \(x\) is \(2y\frac{dy}{dx}\) by the chain rule, and the derivative of \(\ln x\) with respect to \(x\) is \(1/x\). Thus, we obtain the equation \(2y\frac{dy}{dx}=1/x\).

Solve for dy/dx

We are asked to find the derivative of \(y\) with respect to \(x\), which is \(\frac{dy}{dx}\). Therefore, isolate \(\frac{dy}{dx}\) on one side of the equation from step 1: \( \frac{dy}{dx} = \frac{1}{2yx}\).

Substitute the (e, 1) Values into The Equation

For the final step, substitute the given point \((e, 1)\) into the \(\frac{dy}{dx}\) equation from step 2: \( \frac{dy}{dx} = \frac{1}{2*1*e} = \frac{1}{2e} \).