If y=sin4x, At x=0, value of `(dy)/(dx)` is :

Options

A. 0
B. 1
C. -1
D. Infinite

Answer

To find the value of \(\frac{dy}{dx}\) when \(y = \sin(4x)\) at \(x = 0\), we can follow these steps:

### Step 1: Differentiate the function
We start with the function:
\[ y = \sin(4x) \]
To find \(\frac{dy}{dx}\), we differentiate \(y\) with respect to \(x\).

### Step 2: Apply the chain rule
Using the chain rule for differentiation, we know that:
\[ \frac{d}{dx}(\sin(u)) = \cos(u) \cdot \frac{du}{dx} \]
where \(u = 4x\). Therefore, we have:
\[ \frac{dy}{dx} = \cos(4x) \cdot \frac{d}{dx}(4x) \]

### Step 3: Differentiate \(4x\)
The derivative of \(4x\) with respect to \(x\) is:
\[ \frac{d}{dx}(4x) = 4 \]

### Step 4: Combine the results
Now substituting back, we get:
\[ \frac{dy}{dx} = 4 \cdot \cos(4x) \]

### Step 5: Evaluate at \(x = 0\)
Now we need to evaluate \(\frac{dy}{dx}\) at \(x = 0\):
\[ \frac{dy}{dx} \bigg|_{x=0} = 4 \cdot \cos(4 \cdot 0) = 4 \cdot \cos(0) \]

### Step 6: Calculate \(\cos(0)\)
We know that \(\cos(0) = 1\), so:
\[ \frac{dy}{dx} \bigg|_{x=0} = 4 \cdot 1 = 4 \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at \(x = 0\) is:
\[ \boxed{4} \]

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