To differentiate the function \( y = \sin(x^2 + 3) \) with respect to \( x \), we can follow these steps:
### Step 1: Identify the function
Let:
\[
y = \sin(x^2 + 3)
\]
### Step 2: Use the chain rule
Since \( y \) is a function of \( t \) where \( t = x^2 + 3 \), we can apply the chain rule. The chain rule states that:
\[
\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}
\]
### Step 3: Differentiate \( y \) with respect to \( t \)
Now, we differentiate \( y \) with respect to \( t \):
\[
\frac{dy}{dt} = \cos(t) = \cos(x^2 + 3)
\]
### Step 4: Differentiate \( t \) with respect to \( x \)
Next, we differentiate \( t \) with respect to \( x \):
\[
t = x^2 + 3 \implies \frac{dt}{dx} = 2x
\]
### Step 5: Combine the results
Now, substituting back into the chain rule formula:
\[
\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} = \cos(x^2 + 3) \cdot 2x
\]
### Final Result
Thus, the differentiation of \( \sin(x^2 + 3) \) with respect to \( x \) is:
\[
\frac{dy}{dx} = 2x \cos(x^2 + 3)
\]
To differentiate the function \( y = \sin(x^2 + 3) \) with respect to \( x \), we can follow these steps:
### Step 1: Identify the function
Let:
\[
y = \sin(x^2 + 3)
\]
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