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$Given the function f(x)=-(1)/(sqrt(x^(3))), find f^(')(x). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: f^(')(x)=$

Given the function $f(x)=-\frac{1}{\sqrt{x^{3}}}$ , find $f^{\prime}(x)$ . Express your answer in radical form without using negative exponents, simplifying all fractions. $\newline$ Answer: $f^{\prime}(x)=$

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Algebra 2

Csc, sec, and cot of special angles

Full solution

Q. Given the function

f(x)=-\frac{1}{\sqrt{x^{3}}}

, find

f^{\prime}(x)

. Express your answer in radical form without using negative exponents, simplifying all fractions.

\newline

Answer:

f^{\prime}(x)=

Rewrite function: To find the derivative of the function $f(x) = -\frac{1}{\sqrt{x^{3}}}$ , we will use the chain rule and the power rule for derivatives. The function can be rewritten as $f(x) = -x^{-\frac{3}{2}}$ .
Apply power rule: The derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . Applying this rule to $f(x) = -x^{(-3/2)}$ , we get $f'(x) = -(-\frac{3}{2})*x^{(-3/2 - 1)}$ .
Simplify expression: Simplify the expression by multiplying the exponents. This gives us $f'(x) = (\frac{3}{2})\cdot x^{(-\frac{5}{2})}$ .
Convert negative exponent: To express the answer without using negative exponents, we rewrite $x^{-\frac{5}{2}}$ as $\frac{1}{x^{\frac{5}{2}}}$ . So, $f'(x) = \frac{3}{2}\cdot\frac{1}{x^{\frac{5}{2}}}$ .
Express answer: Now, we express $x^{\frac{5}{2}}$ as the square root of $x^5$ , which is $\sqrt{x^5}$ . Therefore, $f'(x) = \frac{3}{2}\left(\frac{1}{\sqrt{x^5}}\right)$ .
Final simplification: Finally, we simplify the fraction by writing $\sqrt{x^5}$ as $\left(x^{\frac{5}{2}}\right)$ . So, $f'(x) = \frac{3}{2}\cdot\frac{1}{x^{\frac{5}{2}}}$ .