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

Answer

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

### Step 1: Identify the function and its derivative
The function we are dealing with is \( y = \cos^{-1}\left(\frac{x}{a}\right) \). We know the derivative of \( \cos^{-1}(u) \) with respect to \( u \) is given by:
\[ \frac{d}{du} \cos^{-1}(u) = -\frac{1}{\sqrt{1 - u^2}} \]
In our case, \( u = \frac{x}{a} \).

### Step 2: Apply the chain rule
To find the derivative of \( y \) with respect to \( x \), we need to apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
where \( u = \frac{x}{a} \).

### Step 3: Compute \( \frac{du}{dx} \)
Now, we calculate \( \frac{du}{dx} \):
\[ u = \frac{x}{a} \implies \frac{du}{dx} = \frac{1}{a} \]

### Step 4: Substitute into the chain rule
Now, substituting \( \frac{dy}{du} \) and \( \frac{du}{dx} \) into the chain rule:
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - \left(\frac{x}{a}\right)^2}} \cdot \frac{1}{a} \]

### 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 by combining the terms under the square root:
\[ \frac{dy}{dx} = -\frac{1}{a \sqrt{\frac{a^2 - x^2}{a^2}}} \]
\[ = -\frac{1}{a} \cdot \frac{\sqrt{a^2}}{\sqrt{a^2 - x^2}} = -\frac{1}{a} \cdot \frac{a}{\sqrt{a^2 - x^2}} = -\frac{1}{\sqrt{a^2 - x^2}} \]

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

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