Differentiate `sin(x^(2)) w.r.t. e^(sin x)`

Answer

To differentiate \( \sin(x^2) \) with respect to \( e^{\sin x} \), we can use the chain rule. We'll denote:

- \( h = \sin(x^2) \)
- \( g = e^{\sin x} \)

We need to find \( \frac{dh}{dg} \). By the chain rule, we have:

\[ \frac{dh}{dg} = \frac{dh/dx}{dg/dx} \]

### Step 1: Differentiate \( h \) with respect to \( x \)

To differentiate \( h = \sin(x^2) \), we use the chain rule:

\[ \frac{dh}{dx} = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x \]

Thus,

\[ \frac{dh}{dx} = 2x \cos(x^2) \]

### Step 2: Differentiate \( g \) with respect to \( x \)

Next, we differentiate \( g = e^{\sin x} \):

\[ \frac{dg}{dx} = e^{\sin x} \cdot \frac{d}{dx}(\sin x) = e^{\sin x} \cdot \cos x \]

### Step 3: Find \( \frac{dh}{dg} \)

Now we can find \( \frac{dh}{dg} \) by dividing \( \frac{dh}{dx} \) by \( \frac{dg}{dx} \):

\[ \frac{dh}{dg} = \frac{\frac{dh}{dx}}{\frac{dg}{dx}} = \frac{2x \cos(x^2)}{e^{\sin x} \cos x} \]

### Final Result

Thus, the derivative of \( \sin(x^2) \) with respect to \( e^{\sin x} \) is:

\[ \frac{dh}{dg} = \frac{2x \cos(x^2)}{e^{\sin x} \cos x} \]

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