Differentiate `cos{sin(x)^(2)}` w.r.t. x.

Answer

To differentiate the function \( y = \cos(\sin(x^2)) \) with respect to \( x \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
Let:
- \( u = \sin(x^2) \) (inner function)
- \( y = \cos(u) \) (outer function)

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

### Step 3: Differentiate the inner function
Next, we differentiate \( u = \sin(x^2) \) with respect to \( x \). Again, we will use the chain rule:
- Let \( v = x^2 \) (inner function of \( u \))
- Then, \( u = \sin(v) \)

The derivative of \( u \) with respect to \( v \) is:
\[ \frac{du}{dv} = \cos(v) \]

Now, differentiate \( v = x^2 \):
\[ \frac{dv}{dx} = 2x \]

Using the chain rule again, we find:
\[ \frac{du}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} = \cos(x^2) \cdot 2x \]

### Step 4: Combine the derivatives using the chain rule
Now we can combine the derivatives:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\sin(u) \cdot \frac{du}{dx} \]

Substituting \( u = \sin(x^2) \):
\[ \frac{dy}{dx} = -\sin(\sin(x^2)) \cdot (2x \cos(x^2)) \]

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

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