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$Given the function f(x)=-(2)/(5sqrtx), find f^(')(x). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: f^(')(x)=$

Given the function $f(x)=-\frac{2}{5 \sqrt{x}}$ , 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{2}{5 \sqrt{x}}

, 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 $f(x) = -\frac{2}{5\sqrt{x}}$ , we will use the power rule for differentiation. The function can be rewritten as $f(x) = -\frac{2}{5x^{\frac{1}{2}}}$ . We will differentiate this with respect to $x$ .
Apply power rule: Using the power rule, the derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . In this case, we have $x^{(1/2)}$ , so the derivative will be $(1/2)*x^{(1/2 - 1)} = (1/2)*x^{(-1/2)}$ .
Use constant multiple rule: Applying the constant multiple rule, the derivative of a constant times a function is the constant times the derivative of the function. Therefore, the derivative of $-(2)/(5x^{(1/2)})$ is $-(2/5)$ times the derivative of $x^{(1/2)}$ , which we found in the previous step.
Combine constant and derivative: Combining the constant and the derivative, we get $f'(x) = -\left(\frac{2}{5}\right) \cdot \left(\frac{1}{2}\right) \cdot x^{-\frac{1}{2}}$ .
Simplify expression: Simplify the expression by multiplying the constants together: $-\left(\frac{2}{5}\right) \times \left(\frac{1}{2}\right) = -\frac{1}{5}$ .
Rewrite in terms of $\sqrt{x}$ : Now, we have $f'(x) = (-\frac{1}{5}) \cdot x^{-\frac{1}{2}}$ . To express this without using negative exponents, we rewrite $x^{-\frac{1}{2}}$ as $\frac{1}{\sqrt{x}}$ .
Final derivative: The final simplified derivative is $f'(x) = (-\frac{1}{5}) \cdot (\frac{1}{\sqrt{x}}) = -\frac{1}{5\sqrt{x}}$ .