if `y = A sin(omega t - kx)`, then the value of `(dy)/(dx)` is

Options

A. `A cos (omega t - kx)`
B. `-A omega cos (omega t - kx)`
C. `AK cos (omega t - kx)`
D. `- AK cos (omega t - kx)`

Answer

To find the value of \(\frac{dy}{dx}\) for the function \(y = A \sin(\omega t - kx)\), we will use the rules of differentiation, particularly the chain rule. Here are the steps to solve the problem:

### Step 1: Identify the function
We have the function:
\[ y = A \sin(\omega t - kx) \]

### Step 2: Differentiate with respect to \(x\)
To find \(\frac{dy}{dx}\), we need to differentiate \(y\) with respect to \(x\). We apply the chain rule here.

### Step 3: Apply the chain rule
The derivative of \(\sin(u)\) with respect to \(u\) is \(\cos(u)\). In our case, \(u = \omega t - kx\). Therefore, we have:
\[ \frac{dy}{dx} = A \cdot \cos(\omega t - kx) \cdot \frac{d}{dx}(\omega t - kx) \]

### Step 4: Differentiate the inner function
Now we differentiate the inner function \(\omega t - kx\) with respect to \(x\):
\[ \frac{d}{dx}(\omega t - kx) = 0 - k = -k \]

### Step 5: Substitute back into the derivative
Now substituting back into our expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = A \cdot \cos(\omega t - kx) \cdot (-k) \]
This simplifies to:
\[ \frac{dy}{dx} = -Ak \cos(\omega t - kx) \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -Ak \cos(\omega t - kx) \]

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