Question

2.1 hw: definition of deri score: 8/9 8/9 answered question 5 find the derivative of: (sqrt10{x}) hint: recall that (sqrtn{x} = x^{\frac{1}{n}}) question help: video submit question jump to answer

Explanation:

Step1: Rewrite the function

Given \(\sqrt[10]{x}\), using the hint \(\sqrt[n]{x}=x^{\frac{1}{n}}\), we rewrite it as \(x^{\frac{1}{10}}\).

Step2: Apply the power rule

The power rule for differentiation is \(\frac{d}{dx}(x^n)=nx^{n - 1}\). Here \(n=\frac{1}{10}\), so we have:
\(\frac{d}{dx}(x^{\frac{1}{10}})=\frac{1}{10}x^{\frac{1}{10}-1}\)

Step3: Simplify the exponent

Simplify \(\frac{1}{10}-1=\frac{1 - 10}{10}=-\frac{9}{10}\). So the derivative is \(\frac{1}{10}x^{-\frac{9}{10}}\) or \(\frac{1}{10x^{\frac{9}{10}}}\) (or \(\frac{1}{10\sqrt[10]{x^9}}\)).

Answer:

\(\frac{1}{10}x^{-\frac{9}{10}}\) (or equivalent forms like \(\frac{1}{10\sqrt[10]{x^9}}\))