Question

Group Activity In Exercises \(43-48,\) use the technique of logarithmic differentiation to find \(d y / d x\) . $$y=x^{\ln x}$$

Short answer

The derivative of the function \(y = x^{\ln x}\) is \(\frac{dy}{dx} = x^{\ln x} * \frac{2 \ln x}{x}\)

Step-by-step solution

Apply the natural logarithm to both sides

Apply the natural logarithm to both sides of the equation \(y = x^{\ln x}\):\[\ln y = \ln(x^{\ln x})\]

Simplify using the rule of logarithms

Use the property of logarithms \(\ln(a^b) = b*\ln(a)\) to simplify the right side of the equation:\[\ln y = \ln x * \ln x\]

Differentiate both sides with respect to x

Now, differentiate both sides with respect to x. Remember that the derivative of \(\ln y\) with respect to x is \(1/y * dy/dx\), and apply the product rule on the right side:\[\frac{1}{y} * \frac{dy}{dx} = \frac{d}{dx}(\ln x * \ln x) = \ln x * \frac{1}{x} + \ln x * \frac{1}{x}\]

Solve for \(dy/dx\)

Now multiply both sides by y to isolate \(dy/dx\) on one side:\[\frac{dy}{dx} = y * (\ln x * \frac{1}{x} + \ln x * \frac{1}{x}) = y * \frac{2 \ln x}{x}\]

Substitute the original equation for y

Finally, substitute the original equation \(y = x^{\ln x}\) back into the equation to get the derivative in terms of x:\[\frac{dy}{dx} = x^{\ln x} * \frac{2 \ln x}{x}\]