Let g(x)=x^((4)/(3)). g^(')(27)=

Apply Power Rule: We need to find the derivative of the function g(x) = x^(4/3). To do this, we will use the power rule for differentiation, which states that if g(x) = x^n, then g'(x) = n*x^(n-1).Calculate Exponent: Applying the power rule to g(x) = x^(4/3), we get g'(x) = (4/3)*x^((4/3)-1). We need to subtract 1 from the exponent (4/3) to apply the power rule correctly.Evaluate Derivative at x=27: Simplifying the exponent, we have (4/3) - 1 = (4/3) - (3/3) = (1/3). So, g'(x) = (4/3)*x^(1/3).Find Cube Root: Now we need to evaluate the derivative at x = 27. So, we substitute x with 27 in the derivative to get g'(27) = (4/3)*(27)^(1/3).Substitute and Simplify: To simplify (27)^(1/3), we need to find the cube root of 27. The cube root of 27 is 3 because 3^3 = 27.Final Answer: Substituting the cube root of 27 into the derivative, we get g'(27) = (4/3)*3.Multiplying (4/3) by 3, we get g'(27) = 4. This is the final answer.

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Q. Let

g(x)=x^{\frac{4}{3}}

\newline

g^{\prime}(27)=

Apply Power Rule: We need to find the derivative of the function $g(x) = x^{\frac{4}{3}}$ . To do this, we will use the power rule for differentiation, which states that if $g(x) = x^n$ , then $g'(x) = n \cdot x^{n-1}$ .
Calculate Exponent: Applying the power rule to $g(x) = x^{\frac{4}{3}}$ , we get $g'(x) = \frac{4}{3}\cdot x^{(\frac{4}{3}-1)}$ . We need to subtract $1$ from the exponent $\frac{4}{3}$ to apply the power rule correctly.
Evaluate Derivative at $x=27$ : Simplifying the exponent, we have $(\frac{4}{3}) - 1 = (\frac{4}{3}) - (\frac{3}{3}) = (\frac{1}{3})$ . So, $g'(x) = (\frac{4}{3})\cdot x^{(\frac{1}{3})}$ .
Find Cube Root: Now we need to evaluate the derivative at $x = 27$ . So, we substitute $x$ with $27$ in the derivative to get $g'(27) = (\frac{4}{3})\cdot(27)^{\frac{1}{3}}$ .
Substitute and Simplify: To simplify $(27)^{1/3}$ , we need to find the cube root of $27$ . The cube root of $27$ is $3$ because $3^3 = 27$ .
Final Answer: Substituting the cube root of $27$ into the derivative, we get $g'(27) = \frac{4}{3}\cdot 3$ .
Final Answer: Substituting the cube root of $27$ into the derivative, we get $g'(27) = (\frac{4}{3})\cdot 3$ . Multiplying $(\frac{4}{3})$ by $3$ , we get $g'(27) = 4$ . This is the final answer.