If `y=sinx` and `x=3t` then `(dy)/(dt)` will be

Options

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

Explanation

To find \(\frac{dy}{dt}\) given that \(y = \sin x\) and \(x = 3t\), we can use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the relationships
We have two equations:
1. \(y = \sin x\)
2. \(x = 3t\)

### Step 2: Differentiate \(y\) with respect to \(t\)
Using the chain rule, we can express \(\frac{dy}{dt}\) as:
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} \]

### Step 3: Find \(\frac{dy}{dx}\)
Differentiate \(y = \sin x\) with respect to \(x\):
\[ \frac{dy}{dx} = \cos x \]

### Step 4: Find \(\frac{dx}{dt}\)
Differentiate \(x = 3t\) with respect to \(t\):
\[ \frac{dx}{dt} = 3 \]

### Step 5: Substitute \(\frac{dy}{dx}\) and \(\frac{dx}{dt}\) into the chain rule expression
Now substitute the values we found into the chain rule expression:
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} = \cos x \cdot 3 \]
Thus, we have:
\[ \frac{dy}{dt} = 3 \cos x \]

### Final Answer
The final expression for \(\frac{dy}{dt}\) is:
\[ \frac{dy}{dt} = 3 \cos x \]