30. If $(\sin x)^y = y^{\cos x}$, then find $\frac{dy}{dx}$. (differentiation, implicit differentiation, logarithmic differentiation, chain rule, trigonometric functions)

Step 1: Apply Natural Logarithm

The given equation involves variables in both the base and the exponent, which suggests using logarithmic differentiation. Taking the natural logarithm on both sides allows us to use the logarithm property $\ln(a^b) = b \ln a$ to bring down the exponents.

Step 2: Differentiate Implicitly

Now we differentiate both sides of the equation with respect to $x$. Since $y$ is a function of $x$, we must use the product rule and chain rule for terms involving $y$.

Step 3: Apply Product Rule on Left Side

For the left side, we apply the product rule $\frac{d}{dx}(uv) = u'v + uv'$ where $u=y$ and $v=\ln(\sin x)$. The derivative of $\ln(\sin x)$ requires the chain rule: $\frac{1}{\sin x} \cdot \cos x$.

Step 4: Apply Product Rule on Right Side

For the right side, we also apply the product rule where $u=\cos x$ and $v=\ln y$. The derivative of $\cos x$ is $-\sin x$. The derivative of $\ln y$ with respect to $x$ requires the chain rule: $\frac{1}{y} \frac{dy}{dx}$.

Step 5: Combine and Rearrange Terms

Now we equate the results from differentiating both sides. We need to isolate $\frac{dy}{dx}$. To do this, we gather all terms containing $\frac{dy}{dx}$ on one side of the equation and move all other terms to the opposite side. We also simplify $\frac{\cos x}{\sin x}$ to $\cot x$.

Step 6: Factor out $\frac{dy}{dx}$

Factor out $\frac{dy}{dx}$ from the terms on the left side of the equation. This makes it easier to solve for $\frac{dy}{dx}$.

Step 7: Solve for $\frac{dy}{dx}$

Finally, divide both sides by the factor multiplied by $\frac{dy}{dx}$ to get the expression for $\frac{dy}{dx}$. To simplify the complex fraction, we multiply the numerator and denominator by $y$ to eliminate the fraction within the denominator.

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