Let f(x)=x^((1)/(2)). f^(')(4)=

Identify Function and Point: Identify the function and the point at which the derivative is to be evaluated. We are given the function f(x) = x^(1/2) and we need to find its derivative at the point x = 4.Differentiate with Power Rule: Differentiate the function with respect to x. To find f'(x), we use the power rule for differentiation, which states that if f(x) = x^n, then f'(x) = n*x^(n-1). Differentiating f(x) = x^(1/2), we get f'(x) = (1/2)*x^((1/2)-1) = (1/2)*x^(-1/2).Simplify Derivative Expression: Simplify the expression for the derivative. Simplifying f'(x) = (1/2)*x^(-1/2), we can write it as f'(x) = (1/2)*(1/x^(1/2)) or f'(x) = 1/(2*sqrt(x)).Evaluate at x = 4: Evaluate the derivative at x = 4. Substitute x = 4 into the derivative f'(x) = 1/(2*sqrt(x)) to find f'(4). f'(4) = 1/(2*sqrt(4)) = 1/(2*2) = 1/4.

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Let $f(x)=x^{\frac{1}{2}}$ . $\newline$ $f^{\prime}(4)=$

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Q. Let

f(x)=x^{\frac{1}{2}}

\newline

f^{\prime}(4)=

Identify Function and Point: Identify the function and the point at which the derivative is to be evaluated. $\newline$ We are given the function $f(x) = x^{\frac{1}{2}}$ and we need to find its derivative at the point $x = 4$ .
Differentiate with Power Rule: Differentiate the function with respect to $x$ . To find $f'(x)$ , we use the power rule for differentiation, which states that if $f(x) = x^n$ , then $f'(x) = n\cdot x^{(n-1)}$ . Differentiating $f(x) = x^{(1/2)}$ , we get $f'(x) = (1/2)\cdot x^{((1/2)-1)} = (1/2)\cdot x^{(-1/2)}$ .
Simplify Derivative Expression: Simplify the expression for the derivative. $\newline$ Simplifying $f'(x) = (\frac{1}{2})*x^{(-\frac{1}{2})}$ , we can write it as $f'(x) = (\frac{1}{2})*(\frac{1}{x^{(\frac{1}{2})}})$ or $f'(x) = \frac{1}{2*\sqrt{x}}$ .
Evaluate at $x = 4$ : Evaluate the derivative at $x = 4$ . $\newline$ Substitute $x = 4$ into the derivative $f'(x) = \frac{1}{2\sqrt{x}}$ to find $f'(4)$ . $\newline$ $f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{2\cdot 2} = \frac{1}{4}$ .