Product and Quotient Rule — Question 4

Question 4 - Solution

We are given: $$f(x) = (x^2 + 3x)\cdot e^{x}$$

Let: $$u(x) = x^2 + 3x, \quad v(x) = e^{x}$$

By the product rule: $$f'(x) = u'(x)v(x) + u(x)v'(x)$$

Differentiate each part: $$u'(x) = 2x + 3, \quad v'(x) = e^{x}$$

Now plug into the formula: $$f'(x) = (2x + 3)e^{x} + (x^2 + 3x)e^{x}$$

Factor out $e^{x}$: $$f'(x) = e^{x} \left[ (2x + 3) + (x^2 + 3x) \right]$$

Simplify inside the brackets: $$(2x + 3) + (x^2 + 3x) = x^2 + 5x + 3$$

Final Answer: $$\boxed{ f'(x) = e^{x}(x^2 + 5x + 3) }$$