What is the derivative of `2^((sinx)^(2))` with respect to `sinx` ?

Options

A. `sinx.2^((sinx)^(2))ln4`
B. `2sinx.2^((sinx)^(2))ln4`
C. `ln(sinx)2^((sinx)^(2))`
D. `2 sinx cosx2^((sinx)^(2))`

Answer

To find the derivative of \( 2^{(\sin x)^2} \) with respect to \( \sin x \), we can follow these steps:

### Step 1: Define the function
Let \( u = 2^{(\sin x)^2} \). We want to find \( \frac{du}{d(\sin x)} \).

### Step 2: Use the chain rule
By the chain rule, we can express the derivative as:
\[ \frac{du}{d(\sin x)} = \frac{du}{dx} \cdot \frac{dx}{d(\sin x)} \]
Since \( \frac{dx}{d(\sin x)} = \cos x \), we can rewrite the equation as:
\[ \frac{du}{d(\sin x)} = \frac{du}{dx} \cdot \cos x \]

### Step 3: Find \( \frac{du}{dx} \)
To find \( \frac{du}{dx} \), we will use the formula for the derivative of an exponential function:
\[ \frac{d}{dx}(a^f) = a^f \ln(a) \cdot \frac{df}{dx} \]
In our case, \( a = 2 \) and \( f = (\sin x)^2 \). Therefore,
\[ \frac{du}{dx} = 2^{(\sin x)^2} \cdot \ln(2) \cdot \frac{d}{dx}((\sin x)^2) \]

### Step 4: Find \( \frac{d}{dx}((\sin x)^2) \)
Using the chain rule again, we have:
\[ \frac{d}{dx}((\sin x)^2) = 2 \sin x \cdot \cos x \]

### Step 5: Substitute back into \( \frac{du}{dx} \)
Now substituting \( \frac{d}{dx}((\sin x)^2) \) into our expression for \( \frac{du}{dx} \):
\[ \frac{du}{dx} = 2^{(\sin x)^2} \cdot \ln(2) \cdot (2 \sin x \cos x) \]
This simplifies to:
\[ \frac{du}{dx} = 2^{(\sin x)^2} \cdot 2 \sin x \cos x \cdot \ln(2) \]

### Step 6: Substitute \( \frac{du}{dx} \) back into the chain rule expression
Now we can substitute \( \frac{du}{dx} \) back into our expression for \( \frac{du}{d(\sin x)} \):
\[ \frac{du}{d(\sin x)} = \left(2^{(\sin x)^2} \cdot 2 \sin x \cos x \cdot \ln(2)\right) \cdot \cos x \]
This simplifies to:
\[ \frac{du}{d(\sin x)} = 2^{(\sin x)^2} \cdot 2 \sin x \cos^2 x \cdot \ln(2) \]

### Final Answer
Thus, the derivative of \( 2^{(\sin x)^2} \) with respect to \( \sin x \) is:
\[ \frac{du}{d(\sin x)} = 2^{(\sin x)^2} \cdot 2 \sin x \cos^2 x \cdot \ln(2) \]