(d)/(dx)(x^((2)/(3)))=

Apply Power Rule: To find the derivative of x^(2/3) with respect to x, we will use the power rule for differentiation. The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).Calculate Derivative: Applying the power rule to x^(2/3), we get the derivative f'(x) = (2/3)*x^((2/3)-1).Simplify Exponent: Subtracting 1 from the exponent (2/3) gives us (2/3) - 3/3, which simplifies to -1/3. So, f'(x) = (2/3)*x^(-1/3).Rewrite Expression: The expression x^(-1/3) can be rewritten as 1/(x^(1/3)). Therefore, the derivative f'(x) = (2/3)*(1/(x^(1/3))).Final Simplified Form: The final simplified form of the derivative is f'(x) = (2/3)*(1/(x^(1/3))) or f'(x) = 2/(3*x^(1/3)).

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

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Algebra 2

Csc, sec, and cot of special angles

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

Apply Power Rule: To find the derivative of $x^{\frac{2}{3}}$ with respect to $x$ , we will use the power rule for differentiation. The power rule states that if $f(x) = x^n$ , then $f'(x) = n \cdot x^{n-1}$ .
Calculate Derivative: Applying the power rule to $x^{\frac{2}{3}}$ , we get the derivative $f'(x) = \frac{2}{3}\cdot x^{\left(\frac{2}{3}-1\right)}$ .
Simplify Exponent: Subtracting $1$ from the exponent $(2/3)$ gives us $(2/3) - 3/3$ , which simplifies to $-1/3$ . So, $f'(x) = (2/3)\cdot x^{-1/3}$ .
Rewrite Expression: The expression $x^{(-1/3)}$ can be rewritten as $1/(x^{(1/3)})$ . Therefore, the derivative $f'(x) = (\frac{2}{3})*(\frac{1}{x^{(1/3)}})$ .
Final Simplified Form: The final simplified form of the derivative is $f'(x) = \frac{2}{3}\cdot\left(\frac{1}{x^{\frac{1}{3}}}\right)$ or $f'(x) = \frac{2}{3x^{\frac{1}{3}}}$ .