Derivative Math Example 5

Solution

1. 1
   Apply the product rule to $x^3 \sin(x)$: $\frac{d}{dx}[x^3 \sin(x)] = 3x^2 \sin(x) + x^3 \cos(x)$.
2. 2
   Apply the quotient rule to $\frac{e^x}{x^2}$: $\frac{d}{dx}\left[\frac{e^x}{x^2}\right] = \frac{e^x \cdot x^2 - e^x \cdot 2x}{x^4} = \frac{e^x(x-2)}{x^3}$.
3. 3
   Combine: $f'(x) = 3x^2 \sin(x) + x^3 \cos(x) - \frac{e^x(x-2)}{x^3}$.
4. 4
   Evaluate at $x = \pi$: $\sin(\pi)=0$, $\cos(\pi)=-1$, so $f'(\pi) = 0 + \pi^3(-1) - \frac{e^\pi(\pi-2)}{\pi^3} = -\pi^3 - \frac{e^\pi(\pi-2)}{\pi^3}$.