`cos ^(-1)(x/a)`

Answer

To find the derivative of the function \( y = \cos^{-1}\left(\frac{x}{a}\right) \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
We start with the function:
\[ y = \cos^{-1}\left(\frac{x}{a}\right) \]

### Step 2: Differentiate using the chain rule
We know that the derivative of \( \cos^{-1}(u) \) with respect to \( u \) is:
\[ \frac{d}{du} \cos^{-1}(u) = -\frac{1}{\sqrt{1 - u^2}} \]
In our case, \( u = \frac{x}{a} \). Therefore, we apply the chain rule:
\[ \frac{dy}{dx} = \frac{d}{du} \cos^{-1}(u) \cdot \frac{du}{dx} \]

### Step 3: Differentiate \( u = \frac{x}{a} \)
Now we need to find \( \frac{du}{dx} \):
\[ u = \frac{x}{a} \implies \frac{du}{dx} = \frac{1}{a} \]

### Step 4: Substitute back into the derivative
Substituting \( u \) and \( \frac{du}{dx} \) back into the derivative:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - \left(\frac{x}{a}\right)^2}} \cdot \frac{1}{a} \]

### Step 5: Simplify the expression
Now we simplify the expression:
\[ \frac{dy}{dx} = -\frac{1}{a \sqrt{1 - \frac{x^2}{a^2}}} \]
This can be rewritten as:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{a^2 - x^2}} \]

### Final Result
Thus, the derivative of the function \( y = \cos^{-1}\left(\frac{x}{a}\right) \) is:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{a^2 - x^2}} \]

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