Differentiate the following w.r.t. x:`sin5x cos 3x`

Answer

To differentiate the function \( y = \sin(5x) \cos(3x) \) with respect to \( x \), we will use the product rule of differentiation and the chain rule. Here is the step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = \sin(5x) \cos(3x) \]

### Step 2: Apply the Product Rule
The product rule states that if \( y = u \cdot v \), then:
\[ \frac{dy}{dx} = u'v + uv' \]
where \( u = \sin(5x) \) and \( v = \cos(3x) \).

### Step 3: Differentiate \( u \) and \( v \)
1. Differentiate \( u = \sin(5x) \):
\[ u' = \cos(5x) \cdot \frac{d}{dx}(5x) = \cos(5x) \cdot 5 = 5\cos(5x) \]

2. Differentiate \( v = \cos(3x) \):
\[ v' = -\sin(3x) \cdot \frac{d}{dx}(3x) = -\sin(3x) \cdot 3 = -3\sin(3x) \]

### Step 4: Substitute \( u, u', v, \) and \( v' \) into the product rule formula
Now we substitute \( u, u', v, \) and \( v' \) into the product rule:
\[ \frac{dy}{dx} = (5\cos(5x))(\cos(3x)) + (\sin(5x))(-3\sin(3x)) \]

### Step 5: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = 5\cos(5x)\cos(3x) - 3\sin(5x)\sin(3x) \]

### Step 6: Use the angle addition formula (optional)
We can use the angle addition formula for sine:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
Thus, we can express the result as:
\[ \frac{dy}{dx} = 5\cos(5x)\cos(3x) - \frac{3}{2} \sin(8x) \]

### Final Answer
The derivative of \( y = \sin(5x) \cos(3x) \) is:
\[ \frac{dy}{dx} = 5\cos(5x)\cos(3x) - 3\sin(5x)\sin(3x) \]

---