Differentiate the following w.r.t.x. `(1)/(sqrt(a^(2)-x^(2)))`

Answer

To differentiate the function \( y = \frac{1}{\sqrt{a^2 - x^2}} \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the Function
We can rewrite the function using exponent notation:
\[ y = (a^2 - x^2)^{-\frac{1}{2}} \]

### Step 2: Differentiate Using the Chain Rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( y \) is:
\[ \frac{dy}{dx} = -\frac{1}{2}(a^2 - x^2)^{-\frac{3}{2}} \cdot \frac{d}{dx}(a^2 - x^2) \]

### Step 3: Differentiate the Inner Function
Now we differentiate the inner function \( a^2 - x^2 \):
\[ \frac{d}{dx}(a^2 - x^2) = 0 - 2x = -2x \]

### Step 4: Substitute Back
Substituting the derivative of the inner function back into the equation gives:
\[ \frac{dy}{dx} = -\frac{1}{2}(a^2 - x^2)^{-\frac{3}{2}} \cdot (-2x) \]

### Step 5: Simplify
Now we simplify the expression:
\[ \frac{dy}{dx} = \frac{x}{(a^2 - x^2)^{\frac{3}{2}}} \]

### Final Result
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{x}{(a^2 - x^2)^{\frac{3}{2}}} \]

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