If `y=cos (sin x )`,then `(dy)/(dx) =`

Options

A. ` (sin x ) sin (sin x ) `
B. ` -(sinx ) sin (sin x) `
C. ` (cos x )sin (sin x) `
D. ` -(cos x )sin ( sin x ) `

Answer

To find the derivative of the function \( y = \cos(\sin x) \), we will use the chain rule of differentiation. Here is the step-by-step solution:

### Step 1: Identify the outer and inner functions
In the function \( y = \cos(\sin x) \), we can identify:
- The outer function \( f(u) = \cos(u) \) where \( u = \sin x \).
- The inner function \( g(x) = \sin x \).

### Step 2: Differentiate the outer function
The derivative of the outer function \( f(u) = \cos(u) \) is:
\[ f'(u) = -\sin(u) \]
So, when we substitute back \( u = \sin x \), we have:
\[ f'(\sin x) = -\sin(\sin x) \]

### Step 3: Differentiate the inner function
Next, we differentiate the inner function \( g(x) = \sin x \):
\[ g'(x) = \cos x \]

### Step 4: Apply the chain rule
According to the chain rule, the derivative of \( y \) with respect to \( x \) is given by:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = -\sin(\sin x) \cdot \cos x \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -\sin(\sin x) \cos x \]

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