`d/(dx) [sin (2x + 3)]`=

Options

A. cos (2x + 3)
B. `2 cos (2x + 3)`
C. `- cos (2x + 3)`
D. `3 cos (2x + 3)`

Answer

To find the derivative of the function \( y = \sin(2x + 3) \), we will apply the chain rule of differentiation. Here’s the step-by-step solution:

### Step 1: Identify the function
Let \( y = \sin(2x + 3) \).

### Step 2: Differentiate using the chain rule
Using the chain rule, the derivative of \( \sin(u) \) where \( u = 2x + 3 \) is given by:
\[ \frac{dy}{dx} = \cos(u) \cdot \frac{du}{dx} \]
where \( u = 2x + 3 \).

### Step 3: Differentiate the inner function
Now, we need to find \( \frac{du}{dx} \):
\[ u = 2x + 3 \]
Differentiating \( u \) with respect to \( x \):
\[ \frac{du}{dx} = 2 \]

### Step 4: Substitute back into the derivative
Now substitute \( u \) and \( \frac{du}{dx} \) back into the derivative:
\[ \frac{dy}{dx} = \cos(2x + 3) \cdot 2 \]

### Step 5: Simplify the expression
Thus, we can write:
\[ \frac{dy}{dx} = 2 \cos(2x + 3) \]

### Final Answer
The derivative of \( \sin(2x + 3) \) with respect to \( x \) is:
\[ \frac{d}{dx} [\sin(2x + 3)] = 2 \cos(2x + 3) \]

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