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What is the value of
(d)/(dx)(sqrtx) at
x=9 ?

What is the value of $\frac{d}{d x}(\sqrt{x})$ at $x=9$ ?

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Find values of derivatives using limits

Full solution

Q. What is the value of

\frac{d}{d x}(\sqrt{x})

at

x=9

?

Identify Function: Identify the function whose derivative we need to find. $\newline$ The function is $f(x) = \sqrt{x}$ , which can also be written as $f(x) = x^{(1/2)}$ .
Apply Power Rule: Apply the power rule for differentiation to find the derivative of $f(x) = x^{\frac{1}{2}}$ . The power rule states that if $f(x) = x^n$ , then $f'(x) = n \cdot x^{n-1}$ . Using this rule, the derivative of $f(x) = x^{\frac{1}{2}}$ is $f'(x) = \frac{1}{2} \cdot x^{\left(\frac{1}{2}-1\right)} = \frac{1}{2} \cdot x^{-\frac{1}{2}}$ .
Simplify Derivative: Simplify the expression for the derivative. $f'(x) = \frac{1}{2}x^{-\frac{1}{2}}$ can be rewritten as $f'(x) = \frac{1}{2}\left(\frac{1}{\sqrt{x}}\right)$ .
Evaluate at $x=9$ : Evaluate the derivative at $x=9$ . $\newline$ Substitute $x$ with $9$ in the expression $f'(x) = \frac{1}{2}\cdot\frac{1}{\sqrt{x}}$ . $\newline$ $f'(9) = \frac{1}{2}\cdot\frac{1}{\sqrt{9}} = \frac{1}{2}\cdot\frac{1}{3} = \frac{1}{6}$ .

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