Differentiate `sin(x^(2)+5)` w.r.t. x.

Answer

To differentiate the function \( y = \sin(x^2 + 5) \) 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:
- Outer function: \( u = \sin(v) \)
- Inner function: \( v = x^2 + 5 \)

### Step 2: Differentiate the outer function
The derivative of \( \sin(v) \) with respect to \( v \) is:
\[ \frac{du}{dv} = \cos(v) \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \( v = x^2 + 5 \):
\[ \frac{dv}{dx} = \frac{d}{dx}(x^2 + 5) = 2x \]

### Step 4: Apply the chain rule
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \cos(v) \cdot 2x \]

### Step 5: Substitute back the inner function
Now substitute \( v = x^2 + 5 \) back into the equation:
\[ \frac{dy}{dx} = \cos(x^2 + 5) \cdot 2x \]

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

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