To differentiate the function \( \sin(x^2 + 3) \) with respect to \( x \), we will apply the chain rule of differentiation. Here’s a step-by-step solution:
### Step 1: Identify the outer and inner functions
The function we need to differentiate is \( \sin(f(x)) \) where \( f(x) = x^2 + 3 \).
### Step 2: Apply the chain rule
According to the chain rule, the derivative of \( \sin(f(x)) \) is given by:
\[
\frac{d}{dx} \sin(f(x)) = \cos(f(x)) \cdot \frac{d}{dx} f(x)
\]
### Step 3: Differentiate the inner function
Next, we need to differentiate \( f(x) = x^2 + 3 \):
\[
\frac{d}{dx} f(x) = \frac{d}{dx} (x^2) + \frac{d}{dx} (3) = 2x + 0 = 2x
\]
### Step 4: Substitute back into the chain rule
Now, substituting \( f(x) \) and its derivative back into the chain rule:
\[
\frac{d}{dx} \sin(x^2 + 3) = \cos(x^2 + 3) \cdot (2x)
\]
### Step 5: Write the final answer
Thus, the derivative of \( \sin(x^2 + 3) \) with respect to \( x \) is:
\[
2x \cos(x^2 + 3)
\]
### Final Answer:
\[
\frac{d}{dx} \sin(x^2 + 3) = 2x \cos(x^2 + 3)
\]
---
To differentiate the function \( \sin(x^2 + 3) \) with respect to \( x \), we will apply the chain rule of differentiation. Here’s a step-by-step solution:
### Step 1: Identify the outer and inner functions
The function we need to differentiate is \( \sin(f(x)) \) where \( f(x) = x^2 + 3 \).
### Step 2: Apply the chain rule
According to the chain rule, the derivative of \( \sin(f(x)) \) is given by:
\[
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