`y= (a^2+x^2)/(sqrt(a^2-x^2))` then dy/dx=

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = \frac{a^2 + x^2}{\sqrt{a^2 - x^2}} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( y = \frac{u}{v} \), then the derivative is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = a^2 + x^2 \) and \( v = \sqrt{a^2 - x^2} \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = a^2 + x^2 \)
- \( v = \sqrt{a^2 - x^2} \)

### Step 2: Differentiate \( u \) and \( v \)
Now, we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

1. **Differentiate \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(a^2 + x^2) = 0 + 2x = 2x \]

2. **Differentiate \( v \)**:
Using the chain rule:
\[ \frac{dv}{dx} = \frac{d}{dx}(\sqrt{a^2 - x^2}) = \frac{1}{2\sqrt{a^2 - x^2}} \cdot (-2x) = -\frac{x}{\sqrt{a^2 - x^2}} \]

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Substituting the values we found:
\[ \frac{dy}{dx} = \frac{\sqrt{a^2 - x^2} \cdot (2x) - (a^2 + x^2) \cdot \left(-\frac{x}{\sqrt{a^2 - x^2}}\right)}{(\sqrt{a^2 - x^2})^2} \]

### Step 4: Simplify the Expression
1. The denominator simplifies to:
\[ (\sqrt{a^2 - x^2})^2 = a^2 - x^2 \]

2. The numerator becomes:
\[ 2x\sqrt{a^2 - x^2} + \frac{x(a^2 + x^2)}{\sqrt{a^2 - x^2}} \]
Combine the terms:
\[ = \frac{2x(a^2 - x^2) + x(a^2 + x^2)}{\sqrt{a^2 - x^2}} \]
\[ = \frac{2xa^2 - 2x^3 + xa^2 + x^3}{\sqrt{a^2 - x^2}} = \frac{3xa^2 - x^3}{\sqrt{a^2 - x^2}} \]

Thus, we have:
\[ \frac{dy}{dx} = \frac{3xa^2 - x^3}{(a^2 - x^2)^{3/2}} \]

### Final Result
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{x(3a^2 - x^2)}{(a^2 - x^2)^{3/2}} \]