derivative of sin 5 x

Answer

To find the derivative of the function \( \sin(5x) \), we can follow these steps:

### Step 1: Identify the function
The given function is:
\[ f(x) = \sin(5x) \]

### Step 2: Use the chain rule
To differentiate \( \sin(5x) \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is:
\[ f'(g(x)) \cdot g'(x) \]

In our case, let:
- \( f(u) = \sin(u) \) where \( u = 5x \)
- Thus, \( g(x) = 5x \)

### Step 3: Differentiate the outer function
The derivative of \( \sin(u) \) with respect to \( u \) is:
\[ \frac{d}{du} \sin(u) = \cos(u) \]

### Step 4: Differentiate the inner function
Now we differentiate \( g(x) = 5x \):
\[ g'(x) = 5 \]

### Step 5: Apply the chain rule
Now we can apply the chain rule:
\[ \frac{d}{dx} \sin(5x) = \cos(5x) \cdot g'(x) = \cos(5x) \cdot 5 \]

### Step 6: Write the final answer
Thus, the derivative of \( \sin(5x) \) is:
\[ \frac{d}{dx} \sin(5x) = 5 \cos(5x) \]

### Final Answer
\[ \frac{d}{dx} \sin(5x) = 5 \cos(5x) \]

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