Question

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

Answer 1

To differentiate \(\frac{5x}{x^5}\) with respect to \(x\), simplify the expression first:

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

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

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

So, the derivative is \(-20x^{-5}\), or equivalently, \(\frac{-20}{x^5}\).