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$Given the function f(x)=(1)/(root(4)(x^(3))), find f^(')(x). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: f^(')(x)=$

Given the function $f(x)=\frac{1}{\sqrt[4]{x^{3}}}$ , 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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Math Problems

Algebra 1

Evaluate piecewise-defined functions

Full solution

Q. Given the function

f(x)=\frac{1}{\sqrt[4]{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: We are given the function $f(x) = \frac{1}{\sqrt[4]{x^{3}}}$ which can be rewritten as $f(x) = x^{-\frac{3}{4}}$ . To find the derivative $f'(x)$ , we will use the power rule for derivatives, which states that if $f(x) = x^n$ , then $f'(x) = n \cdot x^{n-1}$ .
Apply power rule: Applying the power rule to $f(x) = x^{(-3/4)}$ , we get $f'(x) = (-3/4)\cdot x^{(-3/4 - 1)}$ . We subtract $1$ from the exponent to find the new exponent for $x$ .
Simplify derivative: Simplifying the exponent, we have $f'(x) = (-\frac{3}{4})\cdot x^{(-\frac{7}{4})}$ . This is the derivative, but it contains a negative exponent, which we want to avoid.
Avoid negative exponents: To express the derivative without using negative exponents, we rewrite $x^{-7/4}$ as $1/x^{7/4}$ . So, $f'(x) = (-3/4)\cdot(1/x^{7/4})$ .
Express in radical form: Now, we express $x^{\frac{7}{4}}$ in radical form. The expression $x^{\frac{7}{4}}$ is equivalent to the fourth root of $x^7$ , which can be written as $\sqrt[4]{x^7}$ .
Final answer: Therefore, the derivative of the function in radical form without using negative exponents is $f'(x) = \left(-\frac{3}{4}\right)\cdot\left(\frac{1}{\sqrt{x^7}}\right)$ . This is the final answer.