Product and Quotient Rule — Question 5

Question 5 - Solution

We are given: $$f(x) = x^2 e^x \cos x$$

This is a product of three functions: - $u(x) = x^2$ - $v(x) = e^x$ - $w(x) = \cos x$

Use the **product rule** for three functions: $$f'(x) = u'(x) v(x) w(x) + u(x) v'(x) w(x) + u(x) v(x) w'(x)$$

Compute derivatives: - $u'(x) = 2x$ - $v'(x) = e^x$ - $w'(x) = -\sin x$

Now substitute: $$f'(x) = 2x e^x \cos x + x^2 e^x \cos x - x^2 e^x \sin x$$

Factor where possible: $$f'(x) = e^x \left( \cos x (2x + x^2) - x^2 \sin x \right)$$

Final Answer: $$\boxed{ f'(x) = e^x \left[ \cos x (x^2 + 2x) - x^2 \sin x \right] }$$