Question

similar question | all parts showing use the power rule to find the derivative of f(x) = \frac{1}{\sqrt{x^9}}. f(x)=\square

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\frac{1}{\sqrt{x^9}} \) as \( f(x)=x^{-\frac{9}{2}} \) using the property \( \frac{1}{\sqrt[n]{x^m}} = x^{-\frac{m}{n}} \).

Step2: Apply the power rule

The power rule for differentiation is \( \frac{d}{dx}(x^n)=nx^{n - 1} \). Here, \( n=-\frac{9}{2} \), so we have:
\( f'(x)=-\frac{9}{2}x^{-\frac{9}{2}-1} \)

Step3: Simplify the exponent

Simplify the exponent: \( -\frac{9}{2}-1 = -\frac{9}{2}-\frac{2}{2}=-\frac{11}{2} \). So \( f'(x)=-\frac{9}{2}x^{-\frac{11}{2}} \).

Step4: Rewrite in radical form (optional)

We can also rewrite \( x^{-\frac{11}{2}} \) as \( \frac{1}{\sqrt{x^{11}}} \), so \( f'(x)=-\frac{9}{2\sqrt{x^{11}}} \).

Answer:

\( -\frac{9}{2}x^{-\frac{11}{2}} \) (or \( -\frac{9}{2\sqrt{x^{11}}} \))