Differentiate the following w.r.t. x :
`sqrt(1-x^(2)).e^(5x)`

Answer

To differentiate the function \( y = \sqrt{1 - x^2} \cdot e^{5x} \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:

\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step 1: Identify the functions
Let:
- \( u = \sqrt{1 - x^2} \)
- \( v = e^{5x} \)

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

1. **Differentiate \( u \)**:
\[ u = (1 - x^2)^{1/2} \]
Using the chain rule:
\[ \frac{du}{dx} = \frac{1}{2}(1 - x^2)^{-1/2} \cdot (-2x) = \frac{-x}{\sqrt{1 - x^2}} \]

2. **Differentiate \( v \)**:
\[ v = e^{5x} \]
Using the chain rule:
\[ \frac{dv}{dx} = 5e^{5x} \]

### Step 3: Apply the product rule
Now we can apply the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting the values we found:
\[ \frac{dy}{dx} = \sqrt{1 - x^2} \cdot (5e^{5x}) + e^{5x} \cdot \left(\frac{-x}{\sqrt{1 - x^2}}\right) \]

### Step 4: Simplify the expression
Now, we can simplify the expression:
\[ \frac{dy}{dx} = 5\sqrt{1 - x^2} \cdot e^{5x} - \frac{x e^{5x}}{\sqrt{1 - x^2}} \]
We can factor out \( e^{5x} \):
\[ \frac{dy}{dx} = e^{5x} \left( 5\sqrt{1 - x^2} - \frac{x}{\sqrt{1 - x^2}} \right) \]

### Final Answer
Thus, the derivative of \( y = \sqrt{1 - x^2} \cdot e^{5x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = e^{5x} \left( 5\sqrt{1 - x^2} - \frac{x}{\sqrt{1 - x^2}} \right) \]