Differentiate the following w.r.t.x :
`sin^(2)(x^(5))`

Answer

To differentiate the function \( y = \sin^2(x^5) \) with respect to \( x \), we will use the chain rule. Here are the steps:

### Step 1: Identify the outer and inner functions
We can see that \( y = \sin^2(u) \) where \( u = x^5 \).

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( \sin^2(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = 2\sin(u)\cos(u) = \sin(2u) \]

### Step 3: Differentiate the inner function
Now, we differentiate \( u = x^5 \) with respect to \( x \):
\[ \frac{du}{dx} = 5x^4 \]

### Step 4: Apply the chain rule
Now we combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \sin(2u) \cdot 5x^4 \]

### Step 5: Substitute back for \( u \)
Substituting back \( u = x^5 \):
\[ \frac{dy}{dx} = \sin(2x^5) \cdot 5x^4 \]

### Final Answer
Thus, the derivative of \( y = \sin^2(x^5) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 5x^4 \sin(2x^5) \]