Find `dy/dx if y=sin4x`

Answer

To find \( \frac{dy}{dx} \) for the function \( y = \sin(4x) \), we will use the chain rule of differentiation. Here are the steps:

### Step 1: Identify the function and its composition
We have:
\[ y = \sin(4x) \]
This is a composition of two functions: the outer function \( \sin(\theta) \) and the inner function \( \theta = 4x \).

### Step 2: Differentiate the outer function
The derivative of \( \sin(\theta) \) with respect to \( \theta \) is:
\[ \frac{d}{d\theta}(\sin(\theta)) = \cos(\theta) \]
So, applying this to our function:
\[ \frac{dy}{d\theta} = \cos(4x) \]

### Step 3: Differentiate the inner function
Now we need to differentiate the inner function \( \theta = 4x \) with respect to \( x \):
\[ \frac{d\theta}{dx} = 4 \]

### Step 4: Apply the chain rule
According to the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{d\theta}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \cos(4x) \cdot 4 \]

### Step 5: Simplify the expression
Thus, we can write:
\[ \frac{dy}{dx} = 4 \cos(4x) \]

### Final Answer
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 4 \cos(4x) \]

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