Question

The value of the derivative of the function \( y = x e^x \) at \( x = 1 \) is ........... (rounded off to two decimal places).

Hint:

When differentiating a product of two functions, use the product rule: \( \frac{d}{dx}(u \cdot v) = u'v + uv' \).

Answer 1

The given function is: \[ y = x e^x \] Step 1: Differentiate the function using the product rule.
The product rule for differentiation is: \[ \frac{d}{dx}(u \cdot v) = u'v + uv' \] Here, \( u = x \) and \( v = e^x \), so: \[ \frac{dy}{dx} = \frac{d}{dx}(x) \cdot e^x + x \cdot \frac{d}{dx}(e^x) \] Step 2: Find the derivatives: \[ \frac{d}{dx}(x) = 1 \quad \text{and} \quad \frac{d}{dx}(e^x) = e^x \] Thus, the derivative is: \[ \frac{dy}{dx} = 1 \cdot e^x + x \cdot e^x = e^x + x e^x \] Step 3: Now, evaluate the derivative at \( x = 1 \): \[ \frac{dy}{dx} \bigg|_{x=1} = e^1 + 1 \cdot e^1 = e + e = 2e \] Step 4: Substituting \( e \approx 2.718 \): \[ 2e \approx 2 \times 2.718 = 5.436 \] Thus, the value of the derivative at \( x = 1 \) is approximately \( 5.44 \).
Final Answer: \[ \boxed{5.44} \]