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

Answer

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

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

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

### 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 differentiate \( u \) and \( v \):
- \( \frac{du}{dx} = 0 + 2x = 2x \)
- To differentiate \( v \), we use the chain rule:
\[ v = (a^2 - x^2)^{1/2} \]
Thus,
\[ \frac{dv}{dx} = \frac{1}{2}(a^2 - x^2)^{-1/2} \cdot (-2x) = \frac{-x}{\sqrt{a^2 - x^2}} \]

### Step 3: Apply the Quotient Rule
Now we substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:
\[ \frac{dy}{dx} = \frac{\sqrt{a^2 - x^2} \cdot (2x) - (a^2 + x^2) \cdot \left(\frac{-x}{\sqrt{a^2 - x^2}}\right)}{(\sqrt{a^2 - x^2})^2} \]

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

Now substituting back:
\[ \frac{dy}{dx} = \frac{\sqrt{a^2 - x^2} \cdot 2x + (a^2 + x^2) \cdot \frac{x}{\sqrt{a^2 - x^2}}}{a^2 - x^2} \]

### Step 5: Combine the Terms
Combining the terms in the numerator:
\[ \frac{dy}{dx} = \frac{2x\sqrt{a^2 - x^2} + \frac{x(a^2 + x^2)}{\sqrt{a^2 - x^2}}}{a^2 - x^2} \]
This can be rewritten as:
\[ \frac{dy}{dx} = \frac{2x(a^2 - x^2) + x(a^2 + x^2)}{(a^2 - x^2)\sqrt{a^2 - x^2}} \]

### Step 6: Final Simplification
Combining the terms in the numerator:
\[ 2xa^2 - 2x^3 + xa^2 + x^3 = 3xa^2 - x^3 \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{x(3a^2 - x^2)}{(a^2 - x^2)\sqrt{a^2 - x^2}} \]

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