Obtain the differential coefficient of `sin6x.`

Options

A. ` cos 6x`
B. `3 cos 6x`
C. `6 cos x`
D. `6 cos 6x`

Explanation

To find the differential coefficient of the function \( y = \sin(6x) \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

### Step 1: Identify the function
We have the function:
\[ y = \sin(6x) \]

### Step 2: Apply the chain rule
The chain rule states that if you have a composite function \( y = \sin(u) \) where \( u = 6x \), then the derivative \( \frac{dy}{dx} \) can be found using:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate the outer function
First, we differentiate the outer function \( \sin(u) \):
\[ \frac{dy}{du} = \cos(u) \]

### Step 4: Differentiate the inner function
Next, we differentiate the inner function \( u = 6x \):
\[ \frac{du}{dx} = 6 \]

### Step 5: Combine the derivatives
Now, we combine the results from Step 3 and Step 4:
\[ \frac{dy}{dx} = \cos(6x) \cdot 6 \]

### Step 6: Write the final result
Thus, the differential coefficient of \( \sin(6x) \) is:
\[ \frac{dy}{dx} = 6 \cos(6x) \]

### Summary
The differential coefficient of \( \sin(6x) \) is \( 6 \cos(6x) \).

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