Question

1. ( e^{sin(cos x)} = y )

Explanation:

Assuming the problem is to find the derivative of \( y = e^{\sin(\cos x)} \) with respect to \( x \).

Step 1: Identify the outer function

The outer function is \( e^u \) where \( u=\sin(\cos x) \). The derivative of \( e^u \) with respect to \( u \) is \( e^u \).

Step 2: Differentiate the middle function

The middle function is \( \sin(v) \) where \( v = \cos x \). The derivative of \( \sin(v) \) with respect to \( v \) is \( \cos(v) \).

Step 3: Differentiate the inner function

The inner function is \( \cos x \). The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \).

Step 4: Apply the chain rule

The chain rule states that \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dv}\cdot\frac{dv}{dx} \). Substituting the derivatives we found:
\( \frac{dy}{dx}=e^{\sin(\cos x)}\cdot\cos(\cos x)\cdot(-\sin x) \)
Simplify the expression:
\( \frac{dy}{dx}=-e^{\sin(\cos x)}\cos(\cos x)\sin x \)

Answer:

The derivative of \( y = e^{\sin(\cos x)} \) with respect to \( x \) is \( -e^{\sin(\cos x)}\cos(\cos x)\sin x \)