(a) Differentiate $ \left( \frac{5x}{x^5} \right)$ with respect to $x$.

Question

The interval in which $ y = x^2 e^x $ is decreasing is _____

Answer 1

To find the derivative of $\frac{5x}{x^5}$ with respect to $x$, first simplify the expression:

$\frac{5x}{x^5} = 5x^1 \cdot x^{-5} = 5x^{1-5} = 5x^{-4}$

Next, apply the power rule ($\frac{d}{dx}[x^n] = n x^{n-1}$) to differentiate $5x^{-4}$:

$\frac{d}{dx}[5x^{-4}] = 5 \cdot (-4) x^{-4-1} = -20x^{-5}$

The resulting derivative is $-20x^{-5}$, which can also be written as $\frac{-20}{x^5}$.

Solution and Explanation