Let h(x)=x^((1)/(4)). h^(')(16)=

Identify Function and Point: Identify the function and the point at which the derivative is to be evaluated. We are given the function h(x) = x^(1/4) and we need to find the derivative of this function at the point x = 16.Differentiate with Power Rule: Differentiate the function h(x) with respect to x. To find h'(x), we use the power rule for differentiation, which states that if f(x) = x^n, then f'(x) = n*x^(n-1). Applying the power rule to h(x) = x^(1/4), we get: h'(x) = (1/4)*x^(1/4 - 1) h'(x) = (1/4)*x^(-3/4)Evaluate at x = 16: Evaluate the derivative at x = 16. Now we substitute x = 16 into the derivative h'(x) to find h'(16): h'(16) = (1/4)*16^(-3/4)Simplify Expression: Simplify the expression. 16^(-3/4) is the same as 1/(16^(3/4)). Since 16 is 2^4, we can simplify 16^(3/4) as (2^4)^(3/4) = 2^(4*(3/4)) = 2^3 = 8. So, h'(16) = (1/4)*(1/8) h'(16) = 1/32

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Let $h(x)=x^{\frac{1}{4}}$ . $\newline$ $h^{\prime}(16)=$

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Q. Let

h(x)=x^{\frac{1}{4}}

\newline

h^{\prime}(16)=

Identify Function and Point: Identify the function and the point at which the derivative is to be evaluated. $\newline$ We are given the function $h(x) = x^{\frac{1}{4}}$ and we need to find the derivative of this function at the point $x = 16$ .
Differentiate with Power Rule: Differentiate the function $h(x)$ with respect to $x$ . To find $h'(x)$ , we use the power rule for differentiation, which states that if $f(x) = x^n$ , then $f'(x) = n \cdot x^{(n-1)}$ . Applying the power rule to $h(x) = x^{\frac{1}{4}}$ , we get: $h'(x) = \frac{1}{4} \cdot x^{(\frac{1}{4} - 1)}$ $h'(x) = \frac{1}{4} \cdot x^{-\frac{3}{4}}$
Evaluate at $x = 16$ : Evaluate the derivative at $x = 16$ . $\newline$ Now we substitute $x = 16$ into the derivative $h'(x)$ to find $h'(16)$ : $\newline$ $h'(16) = (\frac{1}{4})\cdot16^{(-\frac{3}{4})}$
Simplify Expression: Simplify the expression. $\newline$ $16^{(-3/4)}$ is the same as $1/(16^{(3/4)})$ . Since $16$ is $2^4$ , we can simplify $16^{(3/4)}$ as $(2^4)^{(3/4)} = 2^{(4*(3/4))} = 2^3 = 8$ . $\newline$ So, $h'(16) = (1/4)*(1/8)$ $\newline$ $h'(16) = 1/32$