`y = 5 sin x `

Options

A. `(dy)/(dx) = 5 cos x`
B. `(dy)/(dx) = cos x`
C. `(dy)/(dx) = 25cos x`
D. `(dy)/(dx) = 10cos x`

Answer

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

### Step 1: Identify the function
We start with the function:
\[ y = 5 \sin x \]

### Step 2: Apply the derivative rule
To find the derivative \( \frac{dy}{dx} \), we use the rule for differentiating sine functions. The derivative of \( \sin x \) is \( \cos x \).

### Step 3: Factor out the constant
Since there is a constant (5) multiplied by the sine function, we can factor this constant out when taking the derivative:
\[ \frac{dy}{dx} = 5 \cdot \frac{d}{dx}(\sin x) \]

### Step 4: Differentiate the sine function
Now, we differentiate \( \sin x \):
\[ \frac{d}{dx}(\sin x) = \cos x \]

### Step 5: Combine the results
Now we can substitute back into our derivative:
\[ \frac{dy}{dx} = 5 \cdot \cos x \]

### Final Answer
Thus, the derivative of the function \( y = 5 \sin x \) is:
\[ \frac{dy}{dx} = 5 \cos x \]

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