Question

Group Activity In Exercises \(43-48,\) use the technique of logarithmic differentiation to find \(d y / d x\) . $$y=(\sin x)^{x}, \quad 0

Short answer

The derivative of the given function is \(\frac{dy}{dx} = (\sin x)^x(\ln(\sin x) + \frac{x \cdot \cos(x)}{\sin(x)})\)

Step-by-step solution

Apply Logarithm

Apply the natural logarithm to both sides of the equation to get: \(\ln y = x \ln(\sin x)\)

Differentiate

Now, differentiate both sides with respect to \(x\). This involves product rule on the right side of the equation and the chain rule on the left side. This results in: \(\frac{1}{y}\frac{dy}{dx} = \ln(\sin x) + x \cdot \cos(x)/\sin(x)\)

Rewrite and Multiply Y

Bring \(\frac{dy}{dx}\) by itself on the left side. In doing so, the right side of the equation and the derivative isolated. Multiply both sides by \(y\) (which is \((\sin x)^x\)) to get: \(\frac{dy}{dx} = y(\ln(\sin x) + \frac{x \cdot \cos(x)}{\sin(x)})\) \n Now replace \(y\) with the original function to get the derivative, \(\frac{dy}{dx} = (\sin x)^x(\ln(\sin x) + \frac{x \cdot \cos(x)}{\sin(x)})\)