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$Given the function y=(1)/(6sqrtx), find (dy)/(dx). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: (dy)/(dx)=$

Given the function $y=\frac{1}{6 \sqrt{x}}$ , find $\frac{d y}{d x}$ . Express your answer in radical form without using negative exponents, simplifying all fractions. $\newline$ Answer: $\frac{d y}{d x}=$

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Precalculus

Find trigonometric ratios of special angles

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Q. Given the function

y=\frac{1}{6 \sqrt{x}}

, find

\frac{d y}{d x}

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

\newline

Answer:

\frac{d y}{d x}=

Rewrite function: To find the derivative of the function $y=\frac{1}{6\sqrt{x}}$ with respect to $x$ , we will use the chain rule and the power rule for derivatives. The function can be rewritten as $y = \frac{x^{-\frac{1}{2}}}{6}$ .
Apply power rule: Using the power rule, the derivative of $x^n$ with respect to $x$ is $n\cdot x^{(n-1)}$ , we can differentiate $y$ with respect to $x$ . Here, $n = -\frac{1}{2}$ , so we get $\frac{dy}{dx} = \left(-\frac{1}{2}\right) \cdot x^{\left(-\frac{1}{2} - 1\right)} / 6$ .
Simplify derivative: Simplifying the expression, we get $\frac{dy}{dx} = \left(-\frac{1}{2}\right) \cdot x^{-\frac{3}{2}} / 6 = -\frac{1}{12x^{\frac{3}{2}}}$ .
Express in radical form: Now we express the answer in radical form without using negative exponents. The expression $x^{-\frac{3}{2}}$ is equivalent to $\frac{1}{x^{\frac{3}{2}}}$ or $\frac{1}{\sqrt{x}^3}$ .
Substitute back: Substituting back into our derivative, we get $\frac{dy}{dx} = -\frac{1}{12\sqrt{x}^3}$ . This is the derivative in radical form without negative exponents, and all fractions are simplified.