If ` y= e^(x^(5) e^(x) ) ,then (dy)/(dx) =`

Options

A. ` (5x^(4) +1) e^(x^(5) +x) `
B. ` (5x^(4)- 1) e^(x^(5) + x) `
C. ` (5x^(4) +x) e^(x^(5)+x)`
D. ` (5x^(4) -x) e^(x^(5)+x) `

Answer

To find the derivative of the function \( y = e^{x^5 e^x} \), we will use the chain rule and the product rule of differentiation. Here's the step-by-step solution:

### Step 1: Identify the outer and inner functions
We have:
\[ y = e^{u} \quad \text{where} \quad u = x^5 e^x \]

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = e^u \cdot \frac{du}{dx} \]

### Step 3: Differentiate the inner function \( u \)
Now we need to differentiate \( u = x^5 e^x \). We will use the product rule here:
\[ \frac{du}{dx} = \frac{d}{dx}(x^5) \cdot e^x + x^5 \cdot \frac{d}{dx}(e^x) \]
Calculating each part:
- The derivative of \( x^5 \) is \( 5x^4 \).
- The derivative of \( e^x \) is \( e^x \).

So we have:
\[ \frac{du}{dx} = 5x^4 e^x + x^5 e^x \]

### Step 4: Substitute \( u \) and \( \frac{du}{dx} \) back into the derivative
Now substituting back into the derivative of \( y \):
\[ \frac{dy}{dx} = e^{x^5 e^x} \cdot (5x^4 e^x + x^5 e^x) \]

### Step 5: Factor out common terms
We can factor out \( e^x \) from the expression:
\[ \frac{dy}{dx} = e^{x^5 e^x} \cdot e^x (5x^4 + x^5) \]
This simplifies to:
\[ \frac{dy}{dx} = e^{x^5 e^x + x} (5x^4 + x^5) \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = e^{x^5 e^x + x} (5x^4 + x^5) \]