Differentiate $x^{\sin x}, x>0$ w.r.t. $x$.

Step 1: Introduce a variable and take logarithm

To differentiate a function of the form $f(x)^{g(x)}$, it is often helpful to use logarithmic differentiation. We first set the given expression equal to $y$, and then take the natural logarithm of both sides. This allows us to use the logarithm property $\log(a^b) = b \log a$ to bring the exponent down.

Step 2: Differentiate both sides with respect to x

Now, we differentiate both sides of the equation with respect to $x$. On the left side, we use the chain rule for $\log y$. On the right side, we will need to apply the product rule.

Step 3: Apply the product rule

We apply the product rule, which states that $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$. Here, $u = \sin x$ and $v = \log x$. We know that the derivative of $\log x$ is $\frac{1}{x}$ and the derivative of $\sin x$ is $\cos x$. Substituting these into the product rule gives us the derivative of the right side.

Step 4: Solve for dy/dx

Now we have an expression for $\frac{1}{y} \frac{dy}{dx}$. To find $\frac{dy}{dx}$, we multiply both sides of the equation by $y$.

Step 5: Substitute y back into the equation

Finally, we substitute the original expression for $y$, which was $x^{\sin x}$, back into the equation to get the derivative in terms of $x$ only.

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