Differentiate $\sin ^2 x$ w.r.t. $e^{\cos x}$.

Step 1: Define functions

To differentiate one function with respect to another, we first define each function separately. Let $u = \sin^2 x$ and $v = e^{\cos x}$. We need to find $\frac{du}{dv}$.

Step 2: Differentiate u with respect to x

Now, we differentiate $u$ with respect to $x$. Using the chain rule, the derivative of $\sin^2 x$ is $2 \sin x \cdot \frac{d}{dx}(\sin x) = 2 \sin x \cos x$.

Step 3: Differentiate v with respect to x

Next, we differentiate $v$ with respect to $x$. Using the chain rule, the derivative of $e^{\cos x}$ is $e^{\cos x} \cdot \frac{d}{dx}(\cos x) = e^{\cos x} (-\sin x) = -e^{\cos x} \sin x$.

Step 4: Find du/dv

To find $\frac{du}{dv}$, we use the formula $\frac{du}{dv} = \frac{du/dx}{dv/dx}$. We substitute the expressions we found for $\frac{du}{dx}$ and $\frac{dv}{dx}$.

Step 5: Simplify the expression

We can simplify the expression by canceling out $\sin x$ from the numerator and the denominator, assuming $\sin x \neq 0$. This gives us the final derivative.

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