What is the value of (d)/(dx)(x^((3)/(2))) at x=9 ?

Apply Power Rule: We need to find the derivative of the function f(x) = x^(3/2) with respect to x. To do this, we will use the power rule for differentiation, which states that if f(x) = x^n, then f'(x) = n*x^(n-1).Calculate Derivative: Applying the power rule to our function, we get f'(x) = (3/2)*x^((3/2)-1). Simplifying the exponent, we have f'(x) = (3/2)*x^(1/2).Substitute x=9: Now we need to evaluate the derivative at x=9. We substitute x with 9 in the derivative to get f'(9) = (3/2)*9^(1/2).Simplify Exponent: To simplify 9^(1/2), we recognize that it is the square root of 9, which is 3. So, f'(9) = (3/2)*3.Final Derivative Value: Multiplying (3/2) by 3, we get f'(9) = (3/2)*3 = 9/2 or 4.5.

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What is the value of
(d)/(dx)(x^((3)/(2))) at
x=9 ?

What is the value of $\frac{d}{d x}\left(x^{\frac{3}{2}}\right)$ at $x=9$ ?

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Precalculus

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Q. What is the value of

\frac{d}{d x}\left(x^{\frac{3}{2}}\right)

x=9

Apply Power Rule: We need to find the derivative of the function $f(x) = x^{\frac{3}{2}}$ with respect to $x$ . To do this, we will use the power rule for differentiation, which states that if $f(x) = x^n$ , then $f'(x) = n \cdot x^{n-1}$ .
Calculate Derivative: Applying the power rule to our function, we get $f'(x) = \frac{3}{2}\cdot x^{\left(\frac{3}{2}-1\right)}$ . Simplifying the exponent, we have $f'(x) = \frac{3}{2}\cdot x^{\frac{1}{2}}$ .
Substitute $x=9$ : Now we need to evaluate the derivative at $x=9$ . We substitute $x$ with $9$ in the derivative to get $f'(9) = (\frac{3}{2})\cdot9^{\frac{1}{2}}$ .
Simplify Exponent: To simplify $9^{1/2}$ , we recognize that it is the square root of $9$ , which is $3$ . So, $f'(9) = (3/2)\cdot3$ .
Final Derivative Value: Multiplying $(\frac{3}{2})$ by $3$ , we get $f'(9) = (\frac{3}{2})\cdot3 = \frac{9}{2}$ or $4.5$ .