Question

find the derivative of f(x).

f(x) = sqrt3{\frac{1}{x^{10}}}

write your answer as a constant times a power of x.

f(x) = \boxed{quad}

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\sqrt[3]{\frac{1}{x^{10}}} \) as a power of \( x \). We know that \( \frac{1}{x^{10}} = x^{-10} \) and \( \sqrt[3]{x^n}=x^{\frac{n}{3}} \), so \( f(x)=(x^{-10})^{\frac{1}{3}} = x^{-\frac{10}{3}} \).

Step2: Apply the power rule

The power rule for differentiation is \( \frac{d}{dx}(x^n)=nx^{n - 1} \). Here \( n=-\frac{10}{3} \), so we differentiate \( f(x)=x^{-\frac{10}{3}} \) with respect to \( x \).
Using the power rule, \( f^\prime(x)=-\frac{10}{3}x^{-\frac{10}{3}-1} \).

Step3: Simplify the exponent

Simplify the exponent: \( -\frac{10}{3}-1=-\frac{10}{3}-\frac{3}{3}=-\frac{13}{3} \). So \( f^\prime(x)=-\frac{10}{3}x^{-\frac{13}{3}} \).

Answer:

\( -\frac{10}{3}x^{-\frac{13}{3}} \)